Question 59
Let $ABC$ be a triangle with $AB = 13$, $BC = 14$, and $CA = 15$. We construct isosceles right triangle $ACD$ with $\angle ADC = 90^{\circ}$, where $D$, $B$ are on the same side of line $AC$, and let lines $AD$ and $CB$ meet at $F$. Similarly, we construct isosceles right triangle $BCE$ with $\angle BEC = 90^{\circ}$, where $E$, $A$ are on the same side of line $BC$, and let lines $BE$ and $CA$ meet at $G$. Find $\cos \angle AGF$.
Supplemental figure: Exact triangle with the prescribed right-isosceles constructions and intersections.
Supplemental figure: Exact triangle with the prescribed right-isosceles constructions and intersections.