MathDB

4

Part of 2017 Sharygin Geometry Olympiad

Problems(3)

Congruent triangles after several cuts

Source: Sharygin Finals 2017, Problem 8.4

8/4/2017
Alex dissects a paper triangle into two triangles. Each minute after this he dissects one of obtained triangles into two triangles. After some time (at least one hour) it appeared that all obtained triangles were congruent. Find all initial triangles for which this is possible.
geometrycongruent triangles
Moving circles touch in isosceles triangle

Source: Sharygin Finals 2017, Problem 9.4

8/3/2017
Points MM and KK are chosen on lateral sides AB,ACAB,AC of an isosceles triangle ABCABC and point DD is chosen on BCBC such that AMDKAMDK is a parallelogram. Let the lines MKMK and BCBC meet at point LL, and let X,YX,Y be the intersection points of AB,ACAB,AC with the perpendicular line from DD to BCBC. Prove that the circle with center LL and radius LDLD and the circumcircle of triangle AXYAXY are tangent.
tangencygeometry
8 lines and incircle

Source: Sharygin final round 2017

7/31/2017
Given triangle ABCABC and its incircle ω\omega prove you can use just a ruler and drawing at most 8 lines to construct pointsA′,B′,C′A',B',C' on ω\omega such that A,B′,C′A,B',C' and B,C′,A′B,C',A' and C,A′,B′C,A',B' are collinear.
geometry