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Vietnamese Mathematical Olympiad, 2002

Geometry Difficulty 5.8 AIME, harder Prove it Vietnam

In the plane, let be given an isosceles triangle $ABC$ ( $AB = AC$ ). A variable circle $(O)$ with center $O$ on the line $BC$ , passes through $A$ but does not touch the lines $AB$ , $AC$ . Let $M$ , $N$ be respectively the second points of intersection of the circle $(O)$ with the lines $AB$ , $AC$ . Find the locus of the orthocenter of triangle $AMN$ .

Solution

1st case: $\angle A = 90^\circ$ : the locus is the singleton $\{A\}$ .

2nd case: $\angle A \neq 90^\circ$ : let $D$ be the point symmetric to $A$ with respect to $BC$ , $K$ be the point symmetric to $D$ with respect to $MN$ then the line $HK$ is the image of the line $BC$ under the homothety with center $D$ and ratio $4\sin^2(A/2)$ . Thus the locus of the orthocenter of triangle $AMN$ is $d \setminus \{H_1, H_2\}$ where $d$ is the image of $BC$ under the above mentioned homothety; $H_1, H_2$ are the points on $d$ such that $DH_1 \perp DB$ , $DH_2 \perp DC$ .

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