MathDB

8

Part of 2017 Sharygin Geometry Olympiad

Problems(3)

Median through point on circumcircle of square

Source: Sharygin Finals 2017, Problem 8.8

8/4/2017
Let ABCDABCD be a square, and let PP be a point on the minor arc CDCD of its circumcircle. The lines PA,PBPA, PB meet the diagonals BD,ACBD, AC at points K,LK, L respectively. The points M,NM, N are the projections of K,LK, L respectively to CDCD, and QQ is the common point of lines KNKN and MLML. Prove that PQPQ bisects the segment ABAB.
geometrysquareProjectivecircumcircle
Isotomic points intercepted by common inner tangents

Source: Sharygin Finals 2017, Problem 9.8

8/3/2017
Let AKAK and BLBL be the altitudes of an acute-angled triangle ABCABC, and let ω\omega be the excircle of ABCABC touching side ABAB. The common internal tangents to circles CKLCKL and ω\omega meet ABAB at points PP and QQ. Prove that AP=BQAP =BQ.
Proposed by I.Frolov
geometryhomothety
Special partition of set.

Source: Sharygin 2017 Day 2 Problem 10.8 Grade 10

8/2/2017
10.8 Suppose SS is a set of points in the plane, ∣S∣|S| is even; no three points of SS are collinear. Prove that SS can be partitioned into two sets S1S_1 and S2S_2 so that their convex hulls have equal number of vertices.
geometry