1. Luigi drew on his notebook an isosceles triangle $A B C$ where the sides coming from $A$ are equal. After drawing the internal bisector of angle $\widehat{A B C}$ that intersects side $A C$ at $P$, he noticed that the circle through $B, P, C$ also passes through the midpoint of $A B$. He then wondered what the value of angle $\widehat{B A C}$ was. What is the correct answer?
(A) $30^{\circ}$
(B) $45^{\circ}$
(C) $60^{\circ}$
(D) $90^{\circ}$
(E) $105^{\circ}$
1. The answer is (C). Let $M$ be the midpoint of $A B$. The angles $\widehat{P B M}$ and $\widehat{P C M}$ are congruent because they subtend the same arc. Since the triangle is isosceles, the angles $\widehat{A B C}$ and $\widehat{A C B}$ are equal, and thus, by the difference of equal angles, $\widehat{M C B}=\widehat{P B C}$ and $M$ is also the foot of the angle bisector of $\widehat{A C B}$. Since $P$ is also the midpoint of side $A C$, being the two triangles $M B C$ and $B C P$ congruent, and since $C M$ is both a median and an angle bisector, the triangle $A B C$ is isosceles at $C$, that is, the sides $B C$ and $A C$ are equal. Therefore, $A B C$ is equilateral and the angle in question has a measure of $60^{\circ}$.