MathDB

5

Part of 2023 South East Mathematical Olympiad

Problems(2)

The intersection lies on a circle

Source: 2023 China South-east Mathematical Olympiad Grade 10 P5

7/31/2023
Let ABAB be a chord of the semicircle OO (not the diameter). MM is the midpoint of ABAB, and DD is a point lies on line OMOM (DD is outside semicircle OO). Line ll passes through DD and is parallel to ABAB. P,QP, Q are two points lie on ll and POPO meets semicircle OO at CC. If ∠PCD=∠DMC\angle PCD=\angle DMC, and MM is the orthocentre of △OPQ\triangle OPQ. Prove that the intersection of AQAQ and PBPB lies on semicircle OO.
geometry
concyclic wanted, touchpoints of incircle

Source: 2023 China South East Mathematical Olympiad Grade 11 P5 CSMO

4/6/2024
As shown in the figure, in △ABC\vartriangle ABC, AB>ACAB>AC, the inscribed circle II is tangent to the sides BCBC, CACA, ABAB at points DD, EE, FF respectively, and the straight lines BCBC and EFEF intersect at point KK, DG⊥EFDG \perp EF at point GG, ray IGIG intersects the circumscribed circle of △ABC\vartriangle ABC at point HH. Prove that points HH, GG, DD, KK lie on a circle. https://cdn.artofproblemsolving.com/attachments/5/e/804fb919e9c2f9cf612099e44bad9c75699b2e.png
geometryConcyclicincircle