MathDB

3

Part of 2024 Oral Moscow Geometry Olympiad

Problems(2)

Lines are concur, equilateral triangles, square

Source: Oral Moscow Geometry Olympiad 2024, 8-9.3

9/3/2024
An equilateral triangle ABEABE is built inside the square ABCDABCD on the side ABAB, and an equilateral triangle AFCAFC is built on the diagonal ACAC (DD is inside this triangle). The segment EFEF intersects CDCD at point PP. Prove that the lines APAP, BEBE and CFCF intersect at the same point.
geometry
(CSH) touches the excirlce in right triangle

Source: Oral Moscow Geometry Olympiad 2024, 10-11.3

9/3/2024
The hypotenuse ABAB of a right-angled triangle ABCABC touches the corresponding excircle ω\omega at point TT. Point SS is symmetrical TT relative to the bisector of angle CC, CHCH is the height of the triangle. Prove that the circumcircle of triangle CSHCSH touches the circle ω\omega.
geometryangle bisectorcircumcircle