Question 16
Triangle $ABC$ is inscribed in the circle $C$ and $AA'$, $BB'$, $CC'$ are diameters of this circle. Denote $H_1, H_2, H_3$ the orthocenters of the triangles $A'BC$, $AB'C$, $ABC'$ respectively.
a) Prove that there exists a triangle $PQR$ with sides equal to $[AH_1]$, $[BH_2]$, $[CH_3]$ and compute the ratio $S_{PQR}/S_{ABC}$.
b) Prove that the straight lines $AH_1$, $BH_2$ and $CH_3$ are concurrent.
a) Prove that there exists a triangle $PQR$ with sides equal to $[AH_1]$, $[BH_2]$, $[CH_3]$ and compute the ratio $S_{PQR}/S_{ABC}$.
b) Prove that the straight lines $AH_1$, $BH_2$ and $CH_3$ are concurrent.