Question 523833
Here are some for you to try on your own:


1 (easy). In triangle ABC, the altitude from A onto BC bisects BC. Show that triangle ABC is isosceles.


2 (medium). Prove that the incenter and circumcenter of a triangle are the same point if and only if the triangle is equilateral.


3 (hard). Prove that for any triangle, the circumcenter, orthocenter, and centroid all lie on a line.