Question 52
Let $ABC$ be an acute triangle and $D$ the foot of the angle bisector from $A$. Let $E$ and $F$ be, respectively, the intersection of $AC$ with the circumcircle of $ABD$, and the intersection of $AB$ with the circumcircle of $ACD$. Moreover, let $P$ be the intersection of $BE$ and $CF$.
a. Show that triangle $PBC$ is isosceles.
b. Show that $BD : BE = CD : CF$.
c. Let $F^{\prime}$ be the reflection of $F$ in the midpoint of $BC$. Show that triangle $EBF^{\prime}$ is similar to triangle $ABC$.
a. Show that triangle $PBC$ is isosceles.
b. Show that $BD : BE = CD : CF$.
c. Let $F^{\prime}$ be the reflection of $F$ in the midpoint of $BC$. Show that triangle $EBF^{\prime}$ is similar to triangle $ABC$.