Question 78
The vertices of a right triangle $ABC$ inscribed in a circle divide the circumference into three arcs. The right angle is at $A$, so that the opposite $\operatorname{arc} BC$ is a semicircle while $\operatorname{arc} AB$ and $\operatorname{arc} AC$ are supplementary. To each of the three arcs, we draw a tangent such that its point of tangency is the midpoint of that portion of the tangent intercepted by the extended lines $AB$ and $AC$. More precisely, the point $D$ on $\operatorname{arc} BC$ is the midpoint of the segment joining the points $D'$ and $D''$ where the tangent at $D$ intersects the extended lines $AB$ and $AC$. Similarly for $E$ on arc $AC$ and $F$ on arc $AB$.
Prove that triangle $DEF$ is equilateral.
Prove that triangle $DEF$ is equilateral.