Two prisoners are chained to the walls of a triangular cell. The base is covered with a mirror and a perpendicular wall is separating them. They can each see the opposite corner through a hole. Proof they can see each other through another hole at the base of the perpendicular.
Note that this problem is known as the Blanchet Theorem.
From the purple triangle ABC and the center of its inscribed circle at D, form the three colored circles (BDC), (CDA), and (ADB), with centers F, G, H, respectively, thus forming the yellow triangle FGH. Prove that there is one circle that circumscribes both triangles.
Two semicircles with parallel bases. Show that HF and GE intersect on the perpendicular line to AC above A.
A circle with an inscribed quadrilateral. Prove that ab=cd.
Two lines connected by five triangles. Prove that orange = blue.