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congruent triangles wanted, incircle, 2 excircles and circumcircle related

Source: The Francophone MO Juniors P1

August 10, 2020
congruent trianglescongruentgeometryexcirclescircumcircleFrancophone

Problem Statement

Let ABCABC be a triangle such that AB<ACAB <AC, ω\omega its inscribed circle and Γ\Gamma its circumscribed circle. Let also ωb\omega_b be the excircle relative to vertex BB, then B′B' is the point of tangency between ωb\omega_b and (AC)(AC). Similarly, let the circle ωc\omega_c be the excircle exinscribed relative to vertex CC, then C′C' is the point of tangency between ωc\omega_c and (AB)(AB). Finally, let II be the center of ω\omega and XX the point of Γ\Gamma such that ∠XAI\angle XAI is a right angle. Prove that the triangles XBC′XBC' and XCB′XCB' are congruent.
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