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Romanian District Olympiad 2019 - Grade 9 - Problem 2

Source: Romanian District Olympiad 2019 - Grade 9 - Problem 2

March 18, 2019
geometryVectors

Problem Statement

Let HH be the orthocenter of the acute triangle ABC.ABC. In the plane of the triangle ABCABC we consider a point XX such that the triangle XAHXAH is right and isosceles, having the hypotenuse AH,AH, and BB and XX are on each part of the line AH.AH. Prove that XA→+XC→+XH→=XB→\overrightarrow{XA}+\overrightarrow{XC}+\overrightarrow{XH}=\overrightarrow{XB} if and only if ∠BAC=45∘. \angle BAC=45^{\circ}.
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