MathDB

1

Part of 2020 Francophone Mathematical Olympiad

Problems(2)

congruent triangles wanted, incircle, 2 excircles and circumcircle related

Source: The Francophone MO Juniors P1

8/10/2020
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.
congruent trianglescongruentgeometryexcirclescircumcircleFrancophone
another geometry problem with sharky-devil point

Source: The francophone mathematical olympiads P1

6/27/2020
Let ABCABC be an acute triangle with AC>ABAC>AB, Let DEFDEF be the intouch triangle with D∈(BC)D \in (BC),E∈(AC)E \in (AC),F∈(AB)F \in (AB),, let GG be the intersecttion of the perpendicular from DD to EFEF with ABAB, and X=(ABC)∩(AEF)X=(ABC)\cap (AEF). Prove that B,D,GB,D,G and XX are concylic
geometryincircleFrancophonesharky-devil