Question 146
The acute triangle $ABC$ has the altitude $AD$ and $X$ is a point inside the segment $AD$. The circles $C_1$ and $C_2$ have their centres on the line $BC$ and contain the points $B$ and $X$, respectively $C$ and $X$. The circle $C_1$ meets again the altitude from $B$ of the triangle $ABC$ at $N$ and the side $AB$ at $M$. The circle $C_2$ meets again the altitude from $C$ of the triangle $ABC$ at $P$ and the side $AC$ at $Q$. Let $N^*$ be the intersection of the halfline $AN$ with $C_1$ and $P^*$ be the intersection of the halfline $AP$ with $C_2$.
Prove that $N^*$ and $P^*$ are on the circumcircle of the triangle $ABC$.
Prove that $N^*$ and $P^*$ are on the circumcircle of the triangle $ABC$.