Math Infographic, Geometry Problem 1343: Triangle, Excircles, Circle, Tangent, Tangency Points, Chord, Perpendicular, 90 Degrees, Collinearity. Machu Picchu in the background

The figure below shows a triangle ABC with the excircle O tangent to AC and AB at D and E and excircle Q tangent to AC and AB at G and F. Line GF extended cuts DE at H. Prove that (1) GF is perpendicular to DE; (2) Points O, H, B, and Q are collinear.

Math Infographic, Triangle, Excircles, Circle, Tangent, Tangency Points, Chord, Perpendicular, 90 Degrees, Collinearity. Machu Picchu in the background

Background: Machu Picchu, "Lost City of the Incas"

One of the New Seven Wonders of the World and UNESCO World Heritage Site

"