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5

Part of 2007 Oral Moscow Geometry Olympiad

Problems(2)

3 lines parallel wanted, symmetric of vertices wrt opposite sides

Source: 2007 Oral Moscow Geometry Olympiad grades 8-9 p5

10/18/2020
Given triangle ABCABC. Points A1,B1A_1,B_1 and C1C_1 are symmetric to its vertices with respect to opposite sides. C2C_2 is the intersection point of lines AB1AB_1 and BA1BA_1. PointsA2 A_2 and B2B_2 are defined similarly. Prove that the lines A1A2,B1B2A_1 A_2, B_1 B_2 and C1C2C_1 C_2 are parallel.
(A. Zaslavsky)
geometrySymmetricparallelmidpoint
projections of point P onto lateral faces of pyramid are concyclic

Source: 2007 Oral Moscow Geometry Olympiad grades 10-11 p5

10/19/2020
At the base of the quadrangular pyramid SABCDSABCD lies the quadrangle ABCDABCD. whose diagonals are perpendicular and intersect at point PP, and SPSP is the altitude of the pyramid. Prove that the projections of the point PP onto the lateral faces of the pyramid lie on the same circle.
(A. Zaslavsky)
geometrypyramidConcyclic