MathDB

3

Part of 2022 Francophone Mathematical Olympiad

Problems(2)

P3 froncophone

Source: P3 : froncophone math olympiad junior 2022

5/23/2022
Let △ABC\triangle ABC a triangle, and DD the intersection of the angle bisector of ∠BAC\angle BAC and the perpendicular bisector of ACAC. the line parallel to ACAC passing by the point BB, intersect the line ADAD at XX. the line parallel to CXCX passing by the point BB, intersect ACAC at YY. E=(AYB)∩BXE = (AYB) \cap BX . prove that CC , DD and EE collinear.
geometry
BC//EF wanted, 3 circles related

Source: P3 Francophone Math Olympiad Senior 2022

5/23/2022
Let ABCABC be a triangle and Γ\Gamma its circumcircle. Denote Δ\Delta the tangent at AA to the circle Γ\Gamma. Γ1\Gamma_1 is a circle tangent to the lines Δ\Delta, (AB)(AB) and (BC)(BC), and EE its touchpoint with the line (AB)(AB). Let Γ2\Gamma_2 be a circle tangent to the lines Δ\Delta, (AC)(AC) and (BC)(BC), and FF its touchpoint with the line (AC)(AC). We suppose that EE and FF belong respectively to the segments [AB][AB] and [AC][AC], and that the two circles Γ1\Gamma_1 and Γ2\Gamma_2 lie outside triangle ABCABC. Show that the lines (BC)(BC) and (EF)(EF) are parallel.
geometrycirclesparallel