MathDB

7

Part of 2017 Sharygin Geometry Olympiad

Problems(3)

Arbitrary lines made by regular 13-gons concur

Source: Sharygin Finals 2017, Problem 8.7

8/4/2017
Let A1A2…A13A_1A_2 \dots A_{13} and B1B2…B13B_1B_2 \dots B_{13} be two regular 1313-gons in the plane such that the points B1B_1 and A13A_{13} coincide and lie on the segment A1B13A_1B_{13}, and both polygons lie in the same semiplane with respect to this segment. Prove that the lines A1A9,B13B8A_1A_9, B_{13}B_8 and A8B9A_8B_9 are concurrent.
geometryconcurrency
Maximise number of acute triangles

Source: Sharygin Finals 2017, Problem 9.7

8/3/2017
Let aa and bb be parallel lines with 5050 distinct points marked on aa and 5050 distinct points marked on bb. Find the greatest possible number of acute-angled triangles all of whose vertices are marked.
combinatoricsgeometry
Orthogonal circles in bicentric polygon

Source: Sharygin 2017 Day 2 Problem 10.7 Grade 10

8/2/2017
10.7 A quadrilateral ABCDABCD is circumscribed around the circle ω\omega centered at II and inscribed into the circle Γ\Gamma. The lines AB,CDAB, CD meet at point PP, and the lines BC,ADBC, AD meet at point QQ. Prove that the circles ⊙(PIQ)\odot(PIQ) and Γ\Gamma are orthogonal.
geometry