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#1 h Apr 1, 2016, 7:53 PM • 2 Y

Y by Adventure10, Mango247

In $\triangle{ABC}$ , let $H_a, H_b, H_c$ be the feet of altitudes, and $M_a, M_b, M_c$ be the midpoints of the sides. In triangles $\triangle{H_aM_bM_c}$ , $\triangle{H_bM_aM_c}$ , $\triangle{H_cM_aM_b}$ , let $P_a, P_b, P_c$ be the same points (for example $P_a$ the orthocenter of $\triangle{H_aM_bM_c}$ , $P_b$ the orthocenter of $\triangle{H_bM_aM_c}$ and $P_c$ the orthocenter of $\triangle{H_cM_aM_b}$ . Prove that $\triangle{P_aP_bP_c} \sim \triangle{ABC}$ .

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#2 h Apr 1, 2016, 9:31 PM • 2 Y

Y by Adventure10, Mango247

It is well known that there exists $\odot (H_aH_bH_cM_aM_bM_c)$ for any $\triangle ABC$ referred to as the Nine Point Circle of $\triangle ABC$ , suppose its center is $N$ . Let the point corresponding to $P_a,P_b,P_c$ in $\triangle M_aM_bM_c$ be $P$ . By Midpoint Theorem, $M_bM_c||BC \Rightarrow M_bM_c||H_aM_a \Rightarrow M_bM_cM_aH_a$ is an isosceles trapezoid. Let the $\perp$ -bisector of $M_bM_c$ be $l_a$ , then $N\in l_a$ . Then $\triangle H_aM_bM_c$ is the reflection of $\triangle M_aM_bM_c$ in $l_a$ . Thus $P_a$ is the reflection of $P$ in $l_a \Rightarrow NP_a=NP$ . Similarly, $NP=NP_b=NP_c$ , so $N$ is the circumcenter of $\triangle P_aP_bP_c$ . Also $P_a$ is the reflection of of $P$ in $l_a \Rightarrow \angle PNP_a= 2\angle(l_a,P)$ . Similarly if $l_b$ is the $\perp$ -bisector of $M_aM_c$ , then $\angle PNP_b= 2\angle(l_b,P)$ . Thus $\angle PNP_a+\angle PNP_b = 2\angle(l_a,P)+2\angle(l_b,P) = 2\angle(l_a,l_b) = 2(180^0-\angle M_aM_cM_b) = 360^0-2\angle ACB$ . Since $\angle P_aNP_b+\angle P_aNP +\angle P_bNP=360^0$ , we obtain $\angle P_aNP_b = 2\angle ACB$ using earlier equation. Since $N$ is the circumcenter of $\triangle P_aP_bP_c$ , it follows that $\angle P_aP_cP_b = \frac{1}{2}\angle P_aNP_b = \angle ACB$ . Similarly other angle equalities follow $\Rightarrow \triangle P_aP_bP_c\sim \triangle ABC$ , as required.

By the way, @blaupunkter, can you please tell me the source of this beautiful problem??

This post has been edited 1 time. Last edited by babu2001, Apr 1, 2016, 9:32 PM
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