MathDB

6

Part of 2024 Oral Moscow Geometry Olympiad

Problems(2)

\angle PBA=\angle PCA, AK \perp BC

Source: Oral Moscow Geometry Olympiad 2024, 8-9.6

9/3/2024
Given an acute-angled triangle ABCABC and a point PP inside it such that ∠PBA=∠PCA\angle PBA=\angle PCA. The lines PBPB and PCPC intersect the circumcircles of triangles PCAPCA and PABPAB secondly at points MM and NN, respectively. Let the rays MCMC and NBNB intersect at a point SS, KK is the center of the circumscribed circle of the triangle SMNSMN. Prove that the lines AKAK and BCBC are perpendicular.
geometry
(AN) is touches to (K,KB) and (L,LC)

Source: Oral Moscow Geometry Olympiad 2024, 10-11.6

9/3/2024
An unequal acute-angled triangle ABCABC with an orthocenter HH is given, MM is the midpoint of side BCBC. Points KK and LL lie on a line passing through HH and perpendicular to AMAM such a KBKB and LCLC perpendicular to BCBC. Point NN lies on the line HMHM, and the lines ANAN and AHAH are symmetric with respect to the line AMAM. Prove that a circle with a diameter ANAN touches two circles: centered at KK and with a radius KBKB and with a center LL and radius LCLC.
geometry