MathDB

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Part of 2017 Sharygin Geometry Olympiad

Problems(3)

Parallel line in cyclic kite

Source: Sharygin Finals 2017, Problem 8.1

8/4/2017
Let ABCDABCD be a cyclic quadrilateral with AB=BCAB=BC and AD=CDAD = CD. A point MM lies on the minor arc CDCD of its circumcircle. The lines BMBM and CDCD meet at point PP, the lines AMAM and BDBD meet at point QQ. Prove that PQ∥ACPQ \parallel AC.
geometryProjective
Equilateral triangle with ratio of regular pentagon

Source: Sharygin Finals 2017, Problem 9.1

8/3/2017
Let ABCABC be a regular triangle. The line passing through the midpoint of ABAB and parallel to ACAC meets the minor arc ABAB of the circumcircle at point KK. Prove that the ratio AK:BKAK:BK is equal to the ratio of the side and the diagonal of a regular pentagon.
ratiogeometryGolden Ratio
Midpoint of two circumcentre

Source: Sharygin final round 2017

7/31/2017
If two circles intersect at A,BA,B and common tangents of them intesrsect circles at C,DC,Dif OaO_ais circumcentre of ACDACD and ObO_b is circumcentre of BCDBCD prove ABAB intersects OaObO_aO_b at its midpoint
geometrycircumcircle