User blog comment:Vel!/Call for math facts/@comment-1605058-20140731202912

I don't know if you will be able to see what I posted on IRC chat, so I'm reposting it here:


 * If ABC is a triangle which isn't right, and H is its orthocenter, then ABC, ABH, BCH, CAH have a common nine-point circle. By applying Feuerbach's theorem to these four triangles, we get configuration of 16 circles tangent to the common nine-point circle.


 * Let triangle ABC have orthocenter H distinct from all of its vertices. Every conic passing through A,B,C and H is a rectangular hyperbola. Converse theorem also holds - if three of these points lie on certain rectangular hyperbola, then so does the fourth.


 * All parabolas are similar. Thanks to this fact many facts about parabolas can be verified simply by verifying them for parabola described by equation y=x^2.


 * Encyclopedia of Triangle Centers is a webpage containing description and properties of nearly 6000 points inside a general triangle. In equilateral triangle all these points lie in the same spot.


 * Suppose we have a room inner walls of which are completely covered by mirrors. By using elliptic arcs and reflective properties of ellipse it's possible to make a room in such a way that if we put a point light source in any point, then some part of the room will not be lit. It's unknown if the same is possible in room which has all of its walls flat.