Question 160
Let $ABC$ be an acute, non-isosceles triangle with $H, O, O'$ as its orthocenter, circumcenter, nine-point center, and $D, E, F$ as the midpoints of the segments $BC, CA, AB$, respectively. $P$ is an arbitrary point inside triangle $DEF$. Let $DP, EP, FP$ intersect $(O')$ again at $D', E', F'$, respectively. $A'$ is the reflection of $A$ through $D'$. We define points $B', C'$ similarly.
a. Assume that $PO = PO'$, prove that the circle $(A'B'C')$ passes through $O$.
b. Let $X$ be the reflection of $A'$ with respect to the line $OD$. We define $Y, Z$ similarly. Suppose that $XH, YH, ZH$ intersect $BC, CA, AB$ at $M, N, K$ respectively. Prove that $M, N, K$ are collinear.
a. Assume that $PO = PO'$, prove that the circle $(A'B'C')$ passes through $O$.
b. Let $X$ be the reflection of $A'$ with respect to the line $OD$. We define $Y, Z$ similarly. Suppose that $XH, YH, ZH$ intersect $BC, CA, AB$ at $M, N, K$ respectively. Prove that $M, N, K$ are collinear.