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Rithmatics: Part 7, The Incenter and 9 Point Ellipses

In the last post, we determined that the incenter (the intersection of the angle bisectors of the triangle) is probably the most viable center option for constructing 9-point ellipses. Today we will look at what we can get by using different triangles.

The first thing to note is that for an equilateral triangle the incenter is the same as the orthocenter.  This is because the angle bisectors are also the altitudes of the triangle.  This means that the resulting conic is going to be the 6 point circle we are already familiar with.

When we were working with the orthocenter, right triangles gave us special cases because two of the legs of the triangle were also altitudes.  Since we are using angle bisectors instead of altitudes now, right triangles do not form a special case.  In fact, we will never have a situation where a bisector coincides with a triangle side.  

Another interesting thing to note is that, unlike the orthocenter, the incenter will always be inside the triangle, even if the triangle is obtuse.  This means that obtuse triangles now produce new 9 bind-point patterns.

The most interesting case occurs when we look at isosceles triangles. Here, the angle bisector for the different angle also bisects the base of the triangle.  This means that those two points coincide and we end up with 8-point  ellipses.  Furthermore, the  symmetry of the triangle means that the angle bisector of the unique angle will also end up being an axis of the 8-point ellipse.  If the unique angle is smaller than the two matching angles, it will be the major axis.  If it is larger, it will be the minor axis.  Both possibilities have potential strengths and could be used to design useful defenses.

Next time we will either go off the deep end with a crazy idea I have for Lines of Warding or we will dive into Lines of Vigor. I haven't decided topic I want to write up next.