Question 81
On the sides of an acute $\triangle ABC$ of area $1$ points $A_1 \in BC$, $B_1 \in CA$ and $C_1 \in AB$ are chosen so that
$$
\measuredangle CC_1B = \measuredangle AA_1C = \measuredangle BB_1A = \varphi,
$$
where the angle $\varphi$ is acute. The segments $AA_1$, $BB_1$ and $CC_1$ meet at points $M$, $N$ and $P$.
a) Prove that the circumcenter of $\triangle MNP$ coincides with the orthocenter of $\triangle ABC$.
b) Find $\varphi$, if $S_{MNP} = 2 - \sqrt{3}$.
$$
\measuredangle CC_1B = \measuredangle AA_1C = \measuredangle BB_1A = \varphi,
$$
where the angle $\varphi$ is acute. The segments $AA_1$, $BB_1$ and $CC_1$ meet at points $M$, $N$ and $P$.
a) Prove that the circumcenter of $\triangle MNP$ coincides with the orthocenter of $\triangle ABC$.
b) Find $\varphi$, if $S_{MNP} = 2 - \sqrt{3}$.