You can
put this solution on YOUR website!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.