MathDB

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Part of 2016 Sharygin Geometry Olympiad

Problems(3)

medians of triangle are sidelengths of a right triangle

Source: Sharygin Geometry Olympiad 2016 Final Round problem 1 grade 8

7/22/2018
An altitude AHAH of triangle ABCABC bisects a median BMBM. Prove that the medians of triangle ABMABM are sidelengths of a right-angled triangle.
by Yu.Blinkov
geometryMediansright trianglealtitude
Equal lengths and a parallelogram

Source: Sharygin Geometry Olympiad, Final Round 2016, Problem 1 grade 9

8/4/2016
The diagonals of a parallelogram ABCDABCD meet at point OO. The tangent to the circumcircle of triangle BOCBOC at OO meets ray CBCB at point FF. The circumcircle of triangle FODFOD meets BCBC for the second time at point GG. Prove that AG=ABAG=AB.
geometryAngle Chasingparallelogramcircumcircle
Vertex on radical axis

Source: Sharygin geometry olympiad 2016, grade 10, Final Round, Problem 1.

8/5/2016
A line parallel to the side BCBC of a triangle ABCABC meets the sides ABAB and ACAC at points PP and QQ, respectively. A point MM is chosen inside the triangle APQAPQ. The segments MBMB and MCMC meet the segment PQPQ at points EE and FF, respectively. Let NN be the second intersection point of the circumcircles of the triangles PMFPMF and QMEQME. Prove that the points A,M,NA,M,N are collinear.
geometrycircumcircleSharygin Geometry Olympiad