Question 56
The bisectors of $\angle BAC$, $\angle ABC$ and $\angle ACB$ of $\triangle ABC$ meet its circumcircle at points $A_1$, $B_1$ and $C_1$, respectively. The side $AB$ meets the lines $C_1B_1$ and $C_1A_1$ at points $M$ and $N$, respectively, the side $BC$ meets the lines $A_1C_1$ and $A_1B_1$ at points $P$ and $Q$, respectively, and the side $AC$ meets the lines $B_1A_1$ and $B_1C_1$ at points $R$ and $S$, respectively. Prove that:
a) the altitude of $\triangle CRQ$ through $R$ is equal to the inradius of $\triangle ABC$;
b) the lines $MQ$, $NR$ and $SP$ are concurrent.
a) the altitude of $\triangle CRQ$ through $R$ is equal to the inradius of $\triangle ABC$;
b) the lines $MQ$, $NR$ and $SP$ are concurrent.