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configurational geometry as usual

Source: Indonesia National Math Olympiad 2021 Problem 7 (INAMO 2021/7)

November 9, 2021
geometryif and only ifcircumcircleangle bisector

Problem Statement

Given △ABC\triangle ABC with circumcircle ℓ\ell. Point MM in △ABC\triangle ABC such that AMAM is the angle bisector of ∠BAC\angle BAC. Circle with center MM and radius MBMB intersects ℓ\ell and BCBC at DD and EE respectively, (B≠D,B≠E)(B \not= D, B \not= E). Let PP be the midpoint of arc BCBC in ℓ\ell that didn't have AA. Prove that APAP angle bisector of ∠DPE\angle DPE if and only if ∠B=90∘\angle B = 90^{\circ}.
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