Question 162
The incircle of triangle $ABC$ touches the sides $BC$, $CA$, and $AB$ at $D$, $E$, and $F$ respectively. On the line segments $EF$, $FD$, and $DE$, consider the points $M$, $N$, and $P$ respectively such that the sums $BM + MC$, $CN + NA$, and $AP + PB$ are minimum.
a) Prove that the lines $AM$, $BN$, and $CP$ are concurrent.
b) Prove that $DM$, $EN$ and $FP$ are the altitudes of triangle $DEF$.
a) Prove that the lines $AM$, $BN$, and $CP$ are concurrent.
b) Prove that $DM$, $EN$ and $FP$ are the altitudes of triangle $DEF$.