MathDB

5

Part of 2017 Oral Moscow Geometry Olympiad

Problems(2)

intersecting squares, black triangle area equals to sum of gray areas

Source: 2017 Oral Moscow Geometry Olympiad grades 8-9 p5

7/26/2019
Two squares are arranged as shown. Prove that the area of the black triangle equal to the sum of the gray areas. https://2.bp.blogspot.com/-byhWqNr1ras/XTq-NWusg2I/AAAAAAAAKZA/1sxEZ751v_Evx1ij7K_CGiuZYqCjhm-mQCK4BGAYYCw/s400/Oral%2BSharygin%2B2017%2B8.9%2Bp5.png
geometrySquaresareas
orthocenter, incenter of different triangles and midpoint are collinear

Source: 2017 Oral Moscow Geometry Olympiad grades 10-11 p5

7/25/2019
The inscribed circle of the non-isosceles triangle ABCABC touches sides AB,BCAB, BC and ACAC at points C1,A1C_1, A_1 and B1B_1, respectively. The circumscribed circle of the triangle A1BC1A_1BC_1 intersects the lines B1A1B_1A_1 and B1C1B_1C_1 at the points A0A_0 and C0C_0, respectively. Prove that the orthocenter of triangle A0BC0A_0BC_0, the center of the inscribed circle of triangle ABCABC and the midpoint of the ACAC lie on one straight line.
geometryincentercollinearorthocenter