MathDB

6

Part of 2023 Oral Moscow Geometry Olympiad

Problems(2)

Fixed point geo

Source: Oral Moscow geometry olympiad 2023 8-9.6

4/20/2023
Given a circle Ω\Omega tangent to side ABAB of angle ∠BAC\angle BAC and lying outside this angle. We consider circles ww inscribed in angle BACBAC. The internal tangent of Ω\Omega and ww, different from ABAB, touches ww at a point KK. Let LL be the point of tangency of ww and ACAC. Prove that all such lines KLKL pass through a fixed point without depending on the choice of circle ww.
geometry
Many angle conditions

Source: Oral Moscow geometry olympiad 2023 10-11.6

4/20/2023
Points C1C_1 and C2C_2 lie on side ABAB of triangle ABCABC, where the point C1C_1 belongs to the segment AC2AC_2 and ∠ACC1=∠BCC2\angle ACC_1= \angle BCC_2. On segments CC1CC_1 and CC2CC_2 points A′A' and B′B' are taken such that ∠CAA′=∠CBB′=∠C1CC2\angle CAA'= \angle CBB' = \angle C_1CC_2. Prove that the center of the circle (CA′B′)(CA'B') lies on the perpendicular bisector of the segment ABAB.
geometry