Bulgarian National Olympiad Day 1 Problem 2

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Bulgarian National Olympiad Day 1 Problem 2

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cyclic quadrilateral

geometric transformation

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#1 h Apr 14, 2018, 12:34 PM • 2 Y

Y by Adventure10, Mango247

Let $ABCD$ be a cyclic quadrilateral. Let $H_{1}$ be the orthocentre of triangle $ABC$ . Point $A_{1}$ is the image of $A$ after reflection about $BH_{1}$ . Point $B_{1}$ is the image of of $B$ after reflection about $AH_{1}$ . Let $O_{1}$ be the circumcentre of $(A_{1}B_{1}H_{1})$ . Let $H_{2}$ be the orthocentre of triangle $ABD$ . Point $A_{2}$ is the image of $A$ after reflection about $BH_{2}$ . Point $B_{2}$ is the image of of $B$ after reflection about $AH_{2}$ . Let $O_{2}$ be the circumcentre of $(A_{2}B_{2}H_{2})$ . Lets denote by $\ell_{AB}$ be the line through $O_{1}$ and $O_{2}$ . $\ell_{AD}$ , $\ell_{BC}$ , $\ell_{CD}$ are defined analogously. Let $M=\ell_{AB} \cap \ell_{BC}$ , $N=\ell_{BC} \cap \ell_{CD}$ , $P=\ell_{CD} \cap \ell_{AD}$ , $Q=\ell_{AD} \cap \ell_{AB}$ . Prove that $MNPQ$ is cyclic.

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Dr.Sop

#2 h Apr 14, 2018, 2:12 PM • 2 Y

Y by Adventure10, Mango247

Hint. Prove that the sides of $MNPQ$ are parallel to the sides of $H_1H_2H_3H_4$ .

This post has been edited 1 time. Last edited by Dr.Sop, Apr 15, 2018, 9:13 AM

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Dr.Sop

#3 h Apr 15, 2018, 9:13 AM • 2 Y

Y by Adventure10, Mango247

I used the lemma.

Lemma. Let given $ABC$ with the orthocenter $H$ . Point $A_1$ is the image of $A$ after reflection about $BH$ . Point $B_1$ is the image of $B$ after reflection about $AH$ . Let $O, O_1$ be the circumcenters of $ABC, A_1B_1H$ , then $H, O, O_1$ are collinear and $|HO|/|HO_1| = \sin(\angle BAC)/\sin(180-3\angle BAC)$ .

This post has been edited 1 time. Last edited by Dr.Sop, Apr 15, 2018, 10:43 AM

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