MathNet 2018-part-03 — 181 problems (EN–VI) Question 140

Let $ABC$ be an acute triangle and let $D$ be the foot of the altitude from $A$. On $AD$ lie distinct points $E$ and $F$ such that $|AE| = |BE|$ and $|AF| = |CF|$. A point $T \neq D$ satisfies $\angle BTE = \angle CTF = 90^\circ$. Show that $|TA|^2 = |TB| \cdot |TC|$.



We consider the configuration with the order of the points as in the picture.