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Latest Posts by le-blanc-et-la-noire - Page 2
Otome Isekai Roundup
With the new year, I feel the need to make some kind of year-in-review list. So even though I've largely stopped reading comics (the desire to read ebbs and flows every six month), here are the otoisekai that stuck out to me the most in 2024
Crimson Lady/Resetting Lady
What can I even say about the GOAT? As we crawl towards a conclusion, the despair only grows. Death is the only answer, death must be avoided at all costs.
The Villainess Who is No Better than an Extra Cross-dresses to Be a Love Interest
I never talked about this one, namely because I'm not sure how to describe it without focusing on its wokeness. But it is woke. It's bizarrely, strangely woke, as well as genuinely enjoyable, but I keep getting fixated on all the genre-unusual progressiveness, like
BELIEVABLE female crossdressing in a shojo manga
her older brother is fat, a good person, and nobody ever comments on his weight (this point is the most shocking to me honestly)
Older brother is loudly supportive of his gnc sister and male cousin
Unclear if MC is gay, transmasc, or just doing this out of survival, but it says something that becoming a love interest was her *first* response
Boy the MC bewitched in girl form as a child is thrown off by her handsome appearance as an adult, yet awkwardly asks if she wants help breaking her engagement with a man
I forgot, originally he was the stock "sexist love interest who dials it back for his one true love" type, but this one shatters the mold not by making her the one girl he isn't mean to, but instead having him step back from his feelings and step UP as a genuine, actual, fr ally and continuing their friendship. Insane upgrade.
And it all feels pretty organic! They'll hint at complicated feelings from the people around her without grandstanding or molding medieval-y types into 21st century values
Like I can't emphasize enough how weird it is to see a child drawn like this and they're not evil
Behind Her Highness's Smile
I never talked about this one for the opposite reasons of the above. Um. Extremely horny ethicsplay* thriller about a mentally-challenged princess forced to marry a duke (who, in turn, was forced to marry a mentally-challenged princess). What if you were the abused sexy sensual prisoner princess forcibly married to a smoldering tall dark-haired duke AND you had brain damage.
Let me say that I enjoy this story sincerely and that her issues are not played lightly, but it is absolutely going for eroticism. Like oh nooo, you're not mentally competent enough to consent 😉😉 the duke could do whatever he wants and you're too dumb and doll-like to do anything about it😉😉😉 also your maids roofied you😉did I mention your sick bastard brother-king made you like this? And that he takes immense pleasure in that fact?🤫
But I know what you're thinking. Because I was thinking it. For chapters and chapters. "Surely this is a ruse. They didn't actually write a mentally handicapped female lead, not when it's already so horny. This is just to add to the fucked up atmosphere that feeds into everything because there's no fucking way anyone would keep to this premise. Wow she's really committing to the bit lol, not giving an inch. If I were gullible, I would believe they're actually going to fuck. Wait. Was that guy supposed to represent the writer talking to the reader? Hey?"
Did they? Read to find out :)
*I've always hated describing stories as problematic (positive). Ethicsplay, like the story is fucked up. They know it's fucked up, YOU know it's fucked up, and that's why you're here reading it.
A Splendid Revenge Story of a Super-Dreadnought Cheat Villainess
I don't have a lot to say about this one other than it's a refreshing revenge-centric OI. It's not treading new ground, but it executes the genre's tropes well. The villains are exaggerated caricatures of hubris, brazen leeches who've forgotten whose blood they've been surviving on, each with their own distinctive brand of arrogance introduced at a measured pace to keep the true hero's OPness from getting boring, all with a unique stylistic flair.
Princess revenge stories are frequently derailed by dull romance or the desire to reinvent capitalism, so the fact that Super-Dreadnought commits itself to smiting Lunaria's enemies without straying from the path makes it a high recommend.
Turning the Mad Dog into a Genteel Lord
I realized I don't have enough screenshots to prove my controversial opinion (that this is less puppy dog bf fantasy and more crypto-age reggressor/caretaker right up until they knew it would mess with the overall light and goofy tone), and I can't say those kinds of things without proof. so. I'll save that for another day.
Anyway. Plot: Priestess Diarin, who is so hot I have more screenshots of her than any of the men, has to tame an abused beast-like 6'8 shredded ex-child soldier into a noble. She's the only one who can change him, she's the only one he can be vulnerable with. Middling plot, heavy slapstick/reaction face-based humor, but everyone is sexy and there are mild to moderate sadist-on-the-art-team impies, so. Recommend.
On an unrelated note can tappytoons please stop picking up manwha with good art? Their translation choices make me unreasonably mad.
If they draw shoes like this, please let Lezhin handle it? I can't take the comma stutters and obvious tone neutering.
The Hanged Man by Victor Hugo
i’m sick of these SJWs telling me not to buy bottled water
i propose a new hashtag
#watergate
Sometimes it blows my mind that there are people that don’t wear glasses/contacts. Like they can literally see with no aid. Like they wake up and just be out here seeing. What a wild concept.
Wei Wuxian's Name - The Meaning of Wuxian
C Fans think this is likely the poem behind his name. And after analysing this poem, I think this is also the inspiration behind his character. OMG MXTX YOU’RE A GENIUS 😍.
Before I start, please inform me if you want to use this meta. This analysis is solely mine in every shape and form. There is only 1 Chinese interpretation online and it’s done with Baidu’s MTL. And if I do allow you to use it, PLEASE link it back to me because there’s a very high chance that I will refine this. This is the only translation/meta that I’m particular about so please respect my wishes. 🙏
The term “Wuxian” comes from the Ming Dynasty poem “Living Alone 闲居” by the poet Xiu Ben 徐贲. Sadly, there isn’t much information about the context so I’ve tried to interpret this to the best that I can. There are so many aspects in mdzs/cql that are reflected in his poem.
谢事返丘壑，退耕理田园。
To retire to the hills, to return to farming and prepare the land.
谢事 - To retire and get rid of mundane things, or to die
丘壑 - Hills and ravines. A secluded place in the mountains. A metaphor for somewhere far away.
退耕 - To resign from government service and to return to farming OR to convert land that was used for farming back to other use.
兹心获遂初，稍得洒中悁。
His compassionate heart has attained his wish, and slowly loses itself to anxiety in wine.
兹心 - I think this is related to 慈悲心 which refers to the concept of Compassion in Mahayana Buddhisim. 慈 refers to giving others happiness.
遂初 - This means attained his wish. OR to leave a government post and return to seclusion
悁 - This refers to anger, anxiety and melancholy.
(T/N: 获 and 得 both means to gain, so the poet is emphasising what he has gained)
振策升崇巘，扬舲溯长川。
Whipping the horse (or urging) to ascend the lofty mountain, sailing against the river that never ends
振策 - This term also means to urge or to push someone forward
长川 - It either describes a river OR something that’s never ending.
(T/N: some websites use the word 巘 (mountain), but others use 褵 (a marriage veil). I don’t have enough info, so I’ve gone with the mountain translation.)
惊湍信汨汨，清溜自涓涓。
The water gurgles flows steadily and swiftly, the crystal clear water from the small stream trickled slowly naturally.
(T/N: 汨汨 and 涓涓 are both sounds of the water flowing. 汨汨 refers to gurgling, 涓涓 flows slowly)
新兰艳迟日，密竹曳丛烟。
The new orchid blooms in spring, the dense growth of bamboo gather as they sway like the mist.
迟日 - this could mean late, or spring
烟 - This likely refers to 烟云 (literally translated to mist and cloud), but it also refers to a hermit deep within a faraway mountain.
东馆朝燕坐，西林暮独还。
To meditate in the morning the temple in Chang an, to return from the temple alone at dusk
东馆 - This either refers to the Eastern Temple, or a temple nearby Chang an. The dictionary says it’s 豫中 ‘Yuzhong Temple’ (term was used specifically the Han Dynasty), but I can’t find which temple it is specifically. LOL
西林 - This term does refer to the Xilin temple on Mount Lu. But subsequently it was referred to temples in general. As this a more modern poetry (aka Ming Dynasty), I’ve opted to use ‘temple’ instead.
(T/N: there’s a dawn to dusk imagery in this line, from 朝 (morning) to 暮 (dusk))
朋旧固云旷，山水聊夤缘。
Old friends are resolute and vast as the clouds, and to sit beneath the mountain and rivers to chat about establishing connections.
云 - The imagery of ancient clouds in ancient poetry is uh, not exactly regarded as a good thing. It’s usually related to sadness, or something that’s fickle (ie. a cloud is fickle, it changes shape whenever it wants to, sometimes you see it, and sometimes you don’t.
夤缘 - this means to climb up, establish connections and building relationships)
(T/N: This whole line is interesting and filled with contradictions. 固 refers to something resolute, but 云旷 means vast as the clouds. Of which, clouds symbolise something that’s anything but resolute. And in the second part, 聊 usually refers to a causal chat or even gossip, but 夤缘 refers to using connections to climb higher.)
居喧暂云遣，习静久乃便。
the noise at where I'm at at has stopped for now, dissipating like the wind. And so, the quiet life has persisted for long time.
居 - there are various meanings to this. The most common meaning is to stay/current location so I’ve gone with that
便乃 - this means “hence” or “so”
已幸驻灵药，复能讽瑶编。
It’s already lucky that the magical drug has rooted itself here, and I can memorise the precious books repeatedly
驻 - This is a very interesting word because it’s usually used to mean to garrison or to come to a standstill. It isn’t usually
复 - Repeatedly
讽 - This word usually means to or to mock. But I think it’s used as 讽书 in this case, which is to memorise.
既无羡鱼志，外物非所迁。Even if there isn’t any wishful thinking, the things outside are inhospitable and ever changing
无 - This means no.
羡鱼 - Wishful thinking. This is where Wei Wuxian’s name comes from (MXTX uses 无羡 instead of 羡鱼). This term comes from the idiom “临渊羡鱼，不如退而结网” which literally means, “instead of standing by the water to dream about catching fish, you should go home to make a net to actually catch a fish.” (ie. instead of being envious about another person’s achievements, we should put in the effort) So yes, Wei Wuxian does mean “do not (无) think wishfully and covert over the achievement of others” (羡)
非所 - Inhospitable, where someone cannot live normally
迁 - this means ever changing.
MY ANALYSIS
SO I HAVE SO MUCH FEELS. I believe that MXTX based some parts of Wei Wuxian’s behaviour according to this poem. Allow me to do a line by line analysis.
1st line: “To retire to the hills, to return to farming and prepare the land” - is this why our boy has such a fascination with retiring from his “post” and returning to the simple life? It’s interesting that the terms in this poem repeatedly emphasises on ‘retiring from government.” You could say that wwx’s life, starting from the time that he was a Jiang Clan disciple, and to being a soldier was a form of a post. So the author sets up the stage of what he thinks to “retire” is. And the “hills and the mountains” suggest a secluded far away. IDK about you but it screamed Cloud Recesses to me.
2nd line: “His compassionate heart has attained his wish,” - AGAIN I could scream about this. I feel that Wei Wuxian does embody the concept of “慈悲” in the simple sense of the word. “慈” as in compassionate. He brings joy to people around him and cares for others around him. 悲 as in sadness; he feels the suffering around him and hence he tries to deliver people from it. This is obviously demonstrated when he tries to save the Wen Clan. This line also emphasises that retiring away from all things political is what he wants. And the second part “slowly loses itself to anxiety in wine”, sadly he does this ever so frequently in the novel and almost amplified in CQL.
Line 3 and 4: “Whipping the horse (a metaphor for quickly, almost urgently) to ascend the lofty mountain, sailing against the river that never ends” “The water gurgles flows steadily and swiftly, the crystal clear water from the small stream trickled slowly naturally.” - in Line 3, the first part “ascending the lofty mountain, almost urgently so” reminds me of how he attained his ghost cultivation so quickly over a span of 3 months. And “sailing against the river that never ends” sounds very much like how he was always moving against the criticism of the masses in his first and second life. The lines about the water gurgling and trickling feels that the tide was always against him, be it swiftly or in small trickles.”
Line 5: “The new orchid blooms in spring, the dense growth of bamboo gather as they sway in the mist.” - the imagery of the orchid sounds a shoutout to Jinling. And the dense bamboo dusting in the mist reminds me of the parts where they were in the cold pond. I’m not sure if the CQL directors were equally inspired by this. (I can’t remember if there were bamboos in Cloud Recesses novel wise. 😂😂)
Line 6: “To meditate in the morning the temple in Chang an, to return from the temple alone at dusk” - HMM I don’t have a clear inspiration for this line unfortunately. But it could be a reference to the days spend in Cloud Recesses. The Lan 蓝 was a Buddhism references so it totally fit the bill. (I’ve mentioned it before in Taming Wangxian but if you can’t remember it, well that’s okay, I am to bring it back here eventually)
Line 7: “Old friends are resolute and vast as the clouds, and to sit beneath the mountain and rivers to chat about establishing connections.” - again the comparison of friends with clouds. It does suggest that he has a lot of friends; friends who are as fickle as clouds who aren’t there when the time calls for it. And as for the ones who are “resolute”… well we know who they are. And as for “establishing connections”, this is a common theme in the novel. It actually reminded me of JGY, who was essentially trying to curry favour to promote himself politically.
Line 8-9: “the noise at where I am at has stopped for now, dissipating like the wind. And so, the quiet life has persisted for long time.” “It’s already lucky that the magical drug has rooted itself here, and I can memorise the precious books repeatedly” lmao does the magic drug refer to the incense burner haha 😂😂 the vibe of the last two lines remind me of the end of the novel, whereby the “noise” of society has faded and he’s back to the quiet life. And as for the books that he can read, I’m guessing that’s probably referring to the Lan Family’s library
Line 10: “Even if there isn’t any wishful thinking, the things outside are inhospitable and ever changing” this is WWX and the moral of the story I guess?
(There were about 60 different references and I'm too lazy to link them all lol. They were mostly dictionary links or other poems that used the words in a similar manner. If you want a reference for a particular phrase, please let me know. XD)
In case you missed it, I’ve also analysed Wei Ying previously.
Additional Meta
More MDZS Meta
Meaning of Wei Ying
I have tried for years to discover something, anything, about this card with no success.
happy 110th to the christmas truce!
:3
And looking at all these kinsmen so arrayed, Arjuna, the son of Kunti,
Was overcome by deep compassion; and in despair he said: Krishna, when I see these my own people eager to fight, on the brink,
My limbs grow heavy, and my mouth is parched, my body trembles and my hair bristles,
My bow, Gandiva, falls from my hand, my skin’s on fire, I can no longer stand—my mind is reeling,
I see evil omens, Krishna: nothing good can come from slaughtering one’s own family in battle—I foresee it!
I have no desire for victory, Krishna, or kingship, or pleasures. What should we do with kingship, Govinda? What are pleasures to us? What is life?
The men for whose sake we desire kingship, enjoyment, and pleasures are precisely those drawn up for this battle, having abandoned their lives and riches.
Teachers, fathers, sons, as well as grandfathers, maternal uncles, fathers-in-law, grandsons, brothers-in-law, and kinsmen—
I have no desire to kill them, Madhusudana, though they are killers themselves—no, not for the lordship of the three worlds, let alone the earth!
Where is the joy for us, Janardana, in destroying Dhritarashtra's people? Having killed these murderers, evil would attach itself to us.
It follows, therefore, that we are not required to kill the sons of Dhritarashtra—they are our own kinsmen, and having killed our own people, how could we be happy, Madhava?
And even if, because their minds are overwhelmed by greed, they cannot see the evil incurred by destroying one’s own family, and the degradation involved in the betrayal of a friend,
How can we be so ignorant as not to recoil from this wrong?
~the bhagavad gita 1:27-39, johnson trans
It is customary in my house to offer a bowl of Zyggy* pudding on Christmas Day, as a treat
#EmotionalSupportSpood
*Zyggy, as in Zygoballus sexpunctatus
just had a nice sudden cry thinking about the song "tempus est iocundum" and how much i wish i could meet the anonymous author of that song and many others from the remote past so i could tell them how much i like it, feeling sad that they may never know that even 800 years later someone sings their music every day in the shower and while opening at work and stuff... but i think my tears were tears of gladness too, because maybe through some mystery i will be able to speak to them someday! at times in my past it has felt really certain to me that one day all of time will be gathered up into one perfect moment and we will all get to greet each other forever
The leaves on the wall that are different shades of green and feel leathery and slick and almost oily are the leaves of an ivy, probably Hedera helix.
The expansive bush with arching branches and tight hemispherical clusters of white flowers trailing one by one along each branch is bridal-wreath spirea, Spiraea prunifolia. I love how its leaves resemble fern fronds, or little paws.
it is always an amazing feeling to be out in the world and to remember that you can touch, pick up, and inspect anything for any reason or no reason. almost everything in the world i just look at without touching. but i can touch almost anything
Poetry of Graffiti by chairspoetry
- fubiz
as someone who just finished moby dick & also is rediscovering their love for the mountain goats, I was wondering what your top moby dick tmg songs are. you are the best person i can think of to answer this :)
wow it is truly an honor and a privilege... im limiting this to my personal favorite reading of moby dick which is of course that it is a memory play ishmael puts on ad nauseum because he lives inside the shipwreck and doesn't know how to get out; ahab/whale dynamics would be a whole other project and this already got away from me as it is lol. so, some songs that are about compulsively reliving the day your life ended:
woke up new — i began to talk to myself almost immediately
how to embrace a swamp creature — got out of bed, could not remember my own name / condemned to walk the soil amongst all creatures wild and tame
the slow parts on death metal albums — stock up on gauze in case of accidents / try to keep my story straight
lab rat blues — living water to quench my thirst
going to mexico — i imagined your touch / it was almost too much
keeping house — and when you set the table, set it for two / the ghost on your doorstep has to eat, same as you
maize stalk drinking blood — this is an empty country, and i am the king
historiography — you were warm and that's all i remember
moon over goldsboro — spend all night in the company of ghosts, always wake up alone
almost every door — the moment’s never going to come / when anyone can say that the coast is clear
mobile — lord if you won't keep me safe and warm / then send down the storm, send down the storm
southwestern territory — i try to remember what life was like long ago / but it’s gone, you know
bleed out — i’m gonna make a gigantic mess / but it meant something important i guess
divided sky lane — i will someday learn how to let you go
i could literally keep doing this for hours. but what does everyone else think
it should be "works" both bc the French "oeuvres" has pretty much all of the same meanings, and bc the point of the chapter is faith and works (the question of whether a Christian is justified by right belief or by good actions) and "deeds" is simply not the word we use in English for that concept
I.i.4 Les Œuvres Semblables Aux Paroles
Works Answering Words: Wilbour
Works Resembling Words: Wraxall, Walton
Works Corresponding To Words: Hapgood
Works Matching Words: Denny
Works to Match Words: FMA
He Put His Money Where His Mouth Is: Rose
Deeds to Match Words: Donougher
My copy agrees with blatherby's.
I.i.5 Que Monseigneur Bienvenu Faisait Durer Trop Longtemps Ses Soutanes
How Monseigneur Bienvenu Made His Cassock Last So Long: Wilbour, FMA
Monseigneur's Cassocks Last Too Long: Wraxall
Monseigneur Bienvenu Made His Cassocks Last Too Long: Hapgood
How Monseigneur Bienvenu Made his Cassock last Too Long: Walton, Rose
How Monseigneur Bienvenu Made his Cassocks Last So Long: Denny
Monseigneur Bienvenu Made His Cassocks Last For Too Long: Rose
Unnamed Narrator
from Breakfast at Tiffany's by Truman Capote (1958)
for /-yr/ i like the song La Monture from Notre-Dame de Paris
for the fun-to-say pile: méli-mélo, micmac, assujetissement, eussent été, farfelu
i'm now looking at my list of least favorite french words to pronounce and going "too many r's" for about 40% of them and "skill issue" for most of the rest. some of these are actually very fun to pronounce i just couldn't wrap my tongue around them a year or so ago, but now i can i guess??? so that's very exciting. makes me hope that someday i'll be able to pronounce the rest of them. this is a bit pie in the sky because i really don't see myself ever getting there with procureur du roi but you never know. and luckily the french abolished the monarchy so it's not like i'll ever have to use that phrase in modern conversation.
anyway here are the words i actually love pronouncing now: décaféiné diététicien filleul pneumonie
i now feel normal/neutral about these words that used to be hard for me: automne, condamner douloureux électricité, énergie inférieur, supérieur, etc. itinéraire lourdeur salmonellose sclérose subodorer succincte
words that are definitely within the realm of my current capability but i haven't practiced them enough: bugle hiérarchisation méditerranéen phtisie
words that are still the bane of my existence but i live in hope: [yʁ] plus at least one other r or [y] sound: chirurgie, fourrure, marbrure, moirure, nourriture, ordures, peinturlurer, procureur du roi, prurit, purpurin, sculpture, serrurerie, structure, sulfureux, tournure all words beginning with ur-, hur-, or sur- other difficult sequence of r's and vowels: construire and other -truire verbs; lueur and sueur; utérus too many r's: marbre, martre, meurtre, opprobre, proroger, réfrigérateur, rétrograde, rorqual difficult sequence of vowels and/or semivowels: coopérant, extraordinaire, hémorroïdal, kyrie eleison, météorologique, micro-ordinateur, micro-organisme, mouillure, quatuor, vanillier not pronounced the way i would expect from the spelling: indemne, penta-, punk just hard for some reason: humour
That's Danish; IKEA is Swedish.
the bluetooth chip in my beloved ikea eneby speaker (with the gay pride front cover) decided abruptly to stop connecting to devices today TT__TT i cracked the thing open of course and had a look but unplugging and replugging various cables had no effect; next course of action is probably to try to resolder the bluetooth daughter board (which i HAVE identified, thank you ifixit) except i still haven't unpacked my soldering iron post move and now it's buried in the shed in one of several boxes all of which are behind the woodworking bench and like three bikes. fortunately the 3.5mm jack still works so i THINK my strat is going to be to just fucking plug in one of those bluetooth 3.5mm adapters until i can get my tools unfucked enough to hopefully fix it properly. and of course i can't just get a replacement because ikea has discontinued the eneby because fuck you that's why
I love Golurk's looming, awkward appearance, and how it seems like it's held up by ancient magitek. All of its details contribute to an aesthetic of ruin and age. It seems like such a mournful, protective force. I particularly like the jagged crack in its chest that has been repaired with the same substance as its wristbands. Sometimes I wonder whether it repaired itself, or if another entity did -- and who created it in the first place?
Best Underdog Whose Name Starts With "G" 📂🇬🖋️
Round 2 - Match 4
Our Contestants:
This poll is part of an event that allows the early eliminees from the main tournament have more time in the spotlight!
happy holiday everyone!
On this day, July 27th in 1987, a single was released that would change the world forever.
It's Rick Astley's debut single, Never Gonna Give You Up!
Triangle Tuesday 3: The orthocenter, the Euler line, and orthocentric systems
Previously, we have looked at two different ways to mark a point in a triangle. First, we drew cevians (lines from the vertices) to the midpoints of the sides and found that they all cross at a point, which is the centroid. Then we tried drawing perpendiculars to the sides from the midpoints, and those all met at the circumcenter. And you could do this with any point on the side of a triangle -- draw a cevian to it, or a perpendicular from it, and see what happens.
This time, though, we're going to do both. That is, we're going to work with the cevians that also form perpendiculars to the sides. These are the altitudes, which run from a vertex to the nearest point on the opposite side, called the foot of the altitude. The three altitudes all meet at a point H, and that's the orthocenter. (The letter H has been used to mark the orthocenter since at least the late 19th century. I believe it's from the German Höhenschnittpunkt, "altitude intersection point.") Anyway, let's prove that the orthocenter exists.
Theorem: the three altitudes of a triangle coincide.
Here's a very simple proof that the three altitudes coincide. It relies on the existence of the circumcenter, which we already proved before. Given a triangle ABC, draw a line through A parallel to the opposite side BC. Do the same at B and C. These lines cross at D, E, and F and form the antimedial triangle (in blue).
Then the altitudes of ABC are also the perpendicular bisectors of DEF. We proved before that perpendicular bisectors all meet at a point, therefore altitudes do as well.
That was easy. Let's do it again, but in a different way. It's not quite as simple, but it includes a large bonus.
Theorem: the three altitudes of a triangle coincide at a point colinear with the circumcenter and centroid, and GH = 2 GO.
Let's take triangle ABC, and let F be the midpoint of side AB. Then mark two points that we already know, the circumcenter O and the centroid G. We'll also draw the median (green) and the perpendicular bisector (blue) that run through F, leaving the other ones out to avoid cluttering the picture.
We already know from our look at the centroid that G cuts segment FC at a third of its length, so GC = 2GF. Let's extend segment OG in the direction of G by twice its length out to a point we'll label H, so that GH = 2GO.
Now consider the two triangles GOF and GHC. By construction, their two blue sides are in the ratio 1:2, and the same for their two black sides. They also meet in vertical angles at their common vertex G. So by side-angle-side, the triangles are similar, and it follows that HC is parallel to OF, and therefore perpendicular to AB. So H lies on the altitude from H to side AB.
By analogous construction, we can show that H also lies on the other two altitudes. So not only have we proved that the altitudes coincide, but also that O, G, and H all lie on a line, and furthermore that G is located one third of the way from O to H, in any triangle.
This proof is due to Leonard Euler, and the line OGH is called the Euler line. Not only these three points but many others as well fall on this line, which we will get to later on.
Let's look at some more properties of the orthocenter and the feet of the altitudes. I'm just going to look at the case of an acute triangle for now, and show how this extends to the obtuse case later.
Theorem: two vertices of a triangle and the feet of the altitudes from those vertices are concyclic.
Proof is easy: the two right triangles AHcC and AHaC share segment AC as a hypotenuse. Therefore AC is a diameter of the common circumcircle of AHcC and AHaC (following from Thales's theorem).
(Incidentally, look at the angle formed by the blue segment and the altitude CHc. It subtends the same arc as angle CAHa, so (by the inscribed angle theorem again) they must be equal. That's not a part of this theorem, so just tuck that fact away for a moment.)
Theorem: a vertex, the two adjacent feet of the altitudes, and the orthocenter are concyclic.
Same idea, but now the right triangles are AHcH and AHbH, and AH is the diameter of the common circumcircle.
(And incidentally, look at the angle formed by the new blue line and the altitude CHc. It subtends the same arc as HbAH, which is same angle as CAHa. So those angles must be equal too. Since both angles between a blue line and the altitude CHc are equal to the same thing, they are equal to each other. Again, not a part of this theorem, just something I wanted to note.)
So those are some interesting concyclicities, but now let's look at the pedal triangle of the orthocenter, which is called the orthic triangle.
Oh, hey, it's made up blue lines, just like the ones we were talking about. And we proved that the two longest blue lines make equal angles with the altitude between them. By symmetry, we can prove the same thing about all the angles made by the blue lines. So that means
Theorem: two sides of the orthic triangle make equal angles with the altitude between them.
Another way to say this is that the altitudes are the angle bisectors of the orthic triangle. And I admit that was kind of a roundabout way to introduce the orthic triangle, but I think it makes the proof of this property easier to follow.
Two other properties of the orthic triangle immediately follow from this:
In an acute triangle, the inscribed triangle with the shortest perimeter is the orthic triangle
and
In an acute triangle, the orthic triangle forms a triangular closed path for a beam of light reflected around a triangle
which are two ways of saying the same thing.
But those two properties only hold for acute triangles. What happens to the orthic triangle in an obtuse triangle? Let's push point C downward to make triangle ABC obtuse and see what happens. To make things clear, I'm going to extend the sides of ABC and the altitudes from line segments into lines. Here's the before:
And here's the after:
The orthocenter has moved outside of triangle ABC, and two of the altitudes have their feet on extensions of the sides of ABC rather than on the segments AC and BC. The orthic triangle now extends outside ABC, and certainly isn't the inscribed triangle with the shortest perimeter any more.
But look at it another way. We now have an acute triangle ABH, and the line HHc is an altitude of both the obtuse triangle ABC and the acute triangle ABH. Meanwhile, lines AC and BC have become altitudes of ABH.
So what we have is essentially the same acute triangle with two swaps: point C trades places with H, and Ha trades places with Hb. That means that our two theorems about concyclic points morph into each other as triangle ABC switches between acute and obtuse. Here's an animation to show the process:
And this is why I didn't bother with the obtuse case above -- each theorem of concyclicity is the obtuse case of the other.
So if we can just exchange the orthocenter with one of the vertices, what does this mean for their relationship? If you are given a group of vertices and lines, how can you tell which one is the orthocenter and which one are the vertices? Well, you can't.
Theorem: Given an acute or obtuse triangle ABC and its orthocenter H, A is the orthocenter of triangle BCH, B is the orthocenter of ACH, and C is the orthocenter of ABH.
The proof comes from consulting either of the "before" and "after" figures above. Take any three lines that form a triangle, red or black. The other three lines are then the altitudes of that triangle. The three feet are where a red and black line meet perpendicularly, so they are the same for all four possible triangles, which means all four share the same orthic triangle.
(Of course, if ABC is a right triangle, then we get a degenerate case, as you can see from the gif at the moment when C and H meet.)
Such an arrangement of four points is called an orthocentric system. Of the four points, one is always located inside an acute triangle formed by the other three, and it's conventional to label the interior one H and the others ABC.
Orthocentric systems pop up all over the place in triangles, so expect to see more of them as we go along. Now, let me do one little lemma about altitudes, and then I'll show something cool about orthocentric systems.
Lemma: the segment of an altitude from the orthocenter to a side of the triangle is equal to the extension of the altitude from the side to the circumcircle.
We can show this with just a little shuffling of angle identities. Extend altitude CHc to meet the circumcircle at C'. The angles CAB and CC'B, labeled in red, subtend the same arcs, so they are equal. Triangle ABHb is a right triangle, so angle HbBA, in blue, is complementary to it. The same is true for the right triangle C'BHc, so the two angles labeled in blue are equal. Then by angle-side-angle, triangles BHcH and HHcC' are congruent, and segment HHc = HcC'.
By the same argument, we can show that triangle AHHc is congruent to AC'Hc, which leads us into the next bit.
Theorem: all the circumcircles of the triangles of an orthocentric system are the same size.
The blue triangle has the same circumcircle as triangle ABC. From the foregoing, the blue and green triangles are congruent. Therefore their circumcircles are the same size as well. The same argument works for ACH and BCH.
So here is an orthocentric system with its four circumcircles.
The four circumcenters O, Oa, Ob, and Oc form another orthocentric system, congruent to the first one.
If you found this interesting, please try drawing some of this stuff for yourself! You can use a compass and straightedge, or software such as Geogebra, which I used to make all my drawings. You can try it on the web here or download apps to run on your own computer here.
Triangle Tuesday 2: The circumcenter, pedal triangles, degeneracy, and what even is a triangle anyway?
The circumcenter is almost as simple an idea as the centroid, which we looked at before. To define it, you start the same way. Take triangle ABC, find the midpoints of the sides Ma, Mb, and Mc. Then instead of drawing lines to the midpoints from the vertices, draw perpendicular lines through the midpoints. These lines all coincide at a point O, which is the center of a circle that you can draw through the vertices. The circle is called the circumcircle, and that's why the point is called the circumcenter.
I say almost as simple, but in a sense the circumcenter is simpler than the centroid, because you could easily discover it by accident in the process of simply finding the midpoints. Drawing that perpendicular line, the perpendicular bisector, is the standard way of finding the midpoint of a line segment. It's covered all the way back in Book 1, Proposition 10 of Euclid's Elements, and it's simply this:
So if you find the midpoint of all three sides of a triangle with this method, you've already identified the circumcenter. But that doesn't prove that the perpendicular bisectors always coincide, nor that their point of crossing is the center of the circumcircle. For that, let's return to Euclid (Elements, book 4, proposition 5). Euclid's proof is very straightforward, and leads nicely into something interesting, so we'll follow that, but I will state the theorem differently.
Theorem: the perpendicular bisectors of a triangle coincide and their point of intersection is the center of a circle that meets all three vertices.
Let ABC be a triangle with midpoints of the sides Ma opposite A, similarly for Mb and Mc. Draw perpendiculars to sides AC and BC from their midpoints to meet at point O. Connect three segments from O to A, B, and C.
Consider the two blue triangles. Their sies AMb and CMb are equal, since Mb is the midpoint of AC. They also have OMb in common. Their angles at Mb are right angles, and therefore equal. So they have two sides and one angle the same, making them congruent, and therefore OA = OC.
The same argument applied to the green triangles shows that OB and OC are equal. By transitivity, OA = OB and O is equidistant from the three vertices. The radii of a circle are all equal, so a circle centered at O passing through A also passes through B and C.
Finally, draw a line from O perpendicular to AB. This creates two white triangles with sides OA and OB equal, side OZ in common, and equal right angles at Z. The two triangles are then congruent and the two sides AZ and BZ are equal. So Z is the midpoint Mc, showing that the perpendicular bisectors all meet.
And the same argument works when ABC is obtuse. The circumcenter lands outside the triangle, and in this coloring the white triangles are no longer white, but all the relationships between the segments are the same.
(What Euclid didn't prove is that the perpendicular bisectors of AC and BC do in fact meet somewhere, that is, that they aren't parallel. It's not difficult, but I'm not going to prove that either, at least not yet, for reasons.)
Let's develop another idea. We located the circumcenter by drawing the perpendicular bisectors, but now consider doing this construction in reverse. That is, pick a point, and then draw perpendiculars to the three sides. The intersection of the perpendicular and the side is called the foot of that point with respect to that side. If you do that with with the circumcenter, the feet are of course the midpoints, but you can find the feet for any point.
And if we connect those three feet, we get a triangle. In this case, the medial triangle, which we have seen before. For a point in general, the triangle formed by its feet is called the pedal triangle of that point. ("Pedal" meaning "related to feet," and yes, that is why a lever operated with your foot is also called a pedal.)
So let's draw the pedal triangle for an arbitrary point, move it around, and see what happens. The point is going to sometimes be outside the triangle, but that's all right. With extended sides (dashed lines) we will still be able to draw a perpendicular to find a foot, no matter where the point is.
So there's something interesting -- the three feet become colinear and the pedal triangle flattens out into a straight line when the point is on the circumcircle. Does that always happen?
Looks like it does! So let's prove that. Below is a drawing of the flattened-out pedal triangle of a point on the circumcircle, all labeled up. I've also added a couple dashed lines to make following the proof easier. What we would like to show is that ∠JKP + ∠PKL = 180°.
We're going to extract some information from this drawing based on two facts: a) in a cyclic quadrilateral (meaning it has all vertices on the same circle), opposite angles sum to 180° and b) if two right triangles have the same hypotenuse, the triangles have the same circumcircle. I'm not going to prove either of those here because this post is long enough already, but both of these results follow straightforwardly from the inscribed angle theorem.
Theorem: For a point P on the circumcircle of a triangle ABC, the feet J, K, and L with respect to ABC are colinear.
Okay. PCBA is a cyclic quadrilateral, so
1) ∠BAP + ∠PCB = 180°.
And ∠BAP is the same as ∠LAP, so
2) ∠LAP + ∠PCB = 180°.
The two triangles AKP and ALP are right triangles with the same hypotenuse (the dashed segment AP), so all four points are on the same circle and ALKP is a cyclic quadrilateral. Therefore,
3) ∠LAP + ∠PKL = 180°,
4) ∠PKL = ∠PCB.
Quadrilateral PKCJ is also cyclic (again because of right triangles sharing the same hypotenuse), so
5) ∠JKP = ∠JCP
by the inscribed angle theorem. ∠PCB is supplemental to ∠JCP, so
6) ∠JKP = 180° - ∠PCB
and then combining 4) and 6),
7) ∠JKP + ∠PKL = ∠PCB + (180° - ∠PCB) = 180°,
which means that the pedal triangle of a point on the circumference of a circle is flattened to a line segment. Can we consider such a figure to be a triangle?
Now we can return to Euclid's omission in the existence proof of the circumcircle. Proving that the perpendicular bisectors aren't parallel is equivalent to proving that no two sides of a triangle are parallel, or that the three vertices of a triangle aren't colinear. Euclid didn't do that, but it's pretty simple, so he could have. And then he would simply have said that such an arrangement of line segments isn't a triangle. Modern geometers working with projective geometry can answer differently, and might say that this is a degenerate triangle, but we haven't gotten into that yet.
Let's do one more thing. We can extend the flattened line segment into a line, called the Simson line, after Robert Simson, who never wrote anything about it. It was actually discovered by William Wallace, but not named for him, because that's how things work in math.
The set of all Simson lines from all points on the circumcircle form an envelope in the shape of a deltoid, the Steiner deltoid, named for Jakob Steiner, who for all I can tell was its actual discoverer.
The deltoid is tangent to the sides of the triangle at three points where the Simson line coincides with the sides. I'll have more to say about this lovely deltoid later, but for now, please just enjoy this gif. It took me several hours to figure out how to make it, so if people reading this could spend a collective several hours staring at it, that would be great.
If you found this interesting, please try drawing some of this stuff for yourself! You can use a compass and straightedge, or software such as Geogebra, which I used to make all my drawings. You can try it on the web here or download apps to run on your own computer here.
Let's say there's an online community of people who all have Whatever Syndrome. They talk about all the difficulties and frustrations and issues etc related to Whatever Syndrome. They share advice, they vent, sometimes they just chat and enjoy talking to people who can relate to them properly. Sometimes they make fun relatable observational comedy-style memes about common Whatever Syndrome experiences.
Some of the experiences they make memes about will probably overlap to some extent with the experiences of people who are not on the WS spectrum at all. Let's stipulate (made-up, obviously meaningless numbers incoming) that 10% of the memes they make appeal to a non-WS audience in this way, but 90% are highly specific to the WS niche and won't really be appreciated by outsiders.
In this scenario, the 10% of universally relatable memes will, because they are universally relatable, likely spread far beyond the core WS community. The 90% of niche ones will not (why would they?). From the perspective of someone without WS who doesn't engage with the WS community directly, this will look like 100% of all WS memes seeming to be about things that are just universal human experiences being described as WS-specific experiences for no particular reason. This person might begin to suspect that WS is just a trendy diagnosis that arbitrarily groups completely normal personality traits as a medical issue and that the whole thing is maybe kind of fake. This person is not being unreasonable given the information they have, but for reasons that are hopefully obvious the information they have is very skewed.
On the other hand! If this sort of thing distorts the public perception of what WS is about strongly enough, some people are going to latch onto the relatable memes about it, relate to them (because they're relatable), and wonder if maybe that means they themselves might have WS. This person might do a bit of googling and discover that, in addition to all the relatable stuff they relate to, there are other symptoms that they don't really identify with as much... but then, no one really seems to talk about those things very often, you mostly see people talking about [relatable stuff] when WS comes up, so the latter must be like, the main part, right? So (they think) it can't be too important if the other stuff doesn't apply to me.
[also the whole medical establishment is nightmarishly hard to access and a lot of doctors suck and make diagnoses based on random whims and prejudices, blah blah blah you know all this, the point is that the most obvious solution to "how do I confirm whether I do or don't have a specific medical thing?" is often not reliable.]
Well now, given all of the above... stuff might get confusing huh!
Okay, okay, (you might say), that's all well and good as a toy model of things that might be underlying the discourse you're alluding to, but to what extent is this dynamic actually responsible for what's actually happening? Aha! I have no clue whatsoever, sorry. I'm just the ideas guy.
Big Mood : Bigger Mood :: Bad Analogies : Badger Analogies
this is good and correct. that scene is in fact extremely horny
My toxic trait is finding O'Brien's interrogation of Winston to be extremely horny
This is a good example of what are called "Barnum statements" -- things which are true of almost everyone, but which seem insightful when you read them because they are true of *you*. Furthermore, each result contains many statements (I counted 56 in mine), so that if any one of them matches your current preoccupations or self-concept, you'll forget the ones that didn't apply or only slightly applied.
A fun exercise when you encounter something like this is to tally up all the statements and see how many were really true. In my results, I counted 20 statements that were true about me and 25 that were false.
This doesn't mean that the results are useless, since you may not have thought about your life and emotions in these terms before, and it is still useful to consider e.g. whether you harbor resentment about someone in your life, or whether peace is truly achievable with your current approach. It just means that the usefulness of the results depends entirely on you and your self-knowledge, and the test has no special insight about you.
Quotes from my results under the cut for spoilers.
[disliked olive green] ...inhibiting limitations. Difficult circumstances limit your opportunities for experience and your freedom of action. You feel deprived because you have to do without some of the things that would make life pleasant. You expect far too much understanding for your needs from other people, and as a result, you often feel disappointed. You might ask yourself how much understanding and empathy you extend to others. You would like to be free of your...
I think it's quite clever in the way it mixes different motivations and traits. Here, the main topic is "inhibiting limitations", which makes me think of anxiety and an aversion to breaking rules. But then the next two sentences are about being constrained by non-omnipotence and unable to afford everything you want, which is a universal complaint among finite beings, but not one that seems particularly related to inhibitions. Then, the next two sentences are about wanting others to understand you but not being very understanding of others, which again is very common among humans, but which has even less to do with inhibitions.
As a result, someone whose main problem is any one of {anxiety, poverty, ingratitude} might feel like this paragraph applies to them "uncannily well". And there are 12 such paragraphs! It's no wonder that so many people upthread have felt understood.
where’s that quiz where you choose lke 4 colours u like and 4 u dont and it hands your ass on a plate