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Condition on right angle involving isodynamic points.

Source: Israel Olympic Revenge 2020 SL.

May 3, 2022
geometryangle bisectorapollonius circleNine point center

Problem Statement

Let ABCABC be an acute triangle with AB≠ACAB\neq AC. The angle bisector of ∠BAC\angle BAC intersects with BCBC at a point DD. BE,CFBE,CF are the altitudes of the triangle and Ap1,Ap2Ap_1,Ap_2 are the isodynamic points of triangle ABCABC.Let the AA-median of ABCABC intersect EFEF at TT. Show that the line connecting TT with the nine-point center of ABCABC is perpendicular to BCBC if and only if ∠Ap1DAp2=90∘\angle Ap_1DAp_2=90^\circ.
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