MathDB

3

Part of 2007 Sharygin Geometry Olympiad

Problems(4)

Equal Triangles

Source: 2007 Sharygin Geometry Olympiad. The correspondence round. Grade 8-9. Problem 3

6/7/2016
Segments connecting an inner point of a convex non-equilateral n-gon to its vertices divide the n-gon into n equal triangles. What is the least possible n?
geometryTriangle
diagonals cut a convex in 4 similar triangles, 2 congruent triangles cut also

Source: Sharygin Final 2007 8.3

4/30/2019
The diagonals of a convex quadrilateral dissect it into four similar triangles. Prove that this quadrilateral can also be dissected into two congruent triangles.
geometrysimilar trianglescongruent trianglesconvex quadrilateral
Hexagon concurrency

Source: Sharygin 2007

7/28/2017
Given a hexagon ABCDEFABCDEF such that AB=BCAB=BC, CD=DECD=DE , EF=FAEF=FA and ∠A=∠C=∠E\angle A = \angle C = \angle E Prove that AD,BE,CFAD, BE, CF are concurrent.
geometry
locus of circumcenters of triangles, related to 2intersecting circles

Source: Sharygin 2007 Final 10.3

4/30/2019
Given two circles intersecting at points PP and QQ. Let C be an arbitrary point distinct from PP and QQ on the former circle. Let lines CPCP and CQCQ intersect again the latter circle at points A and B, respectively. Determine the locus of the circumcenters of triangles ABCABC.
geometryCircumcenterLocusLocus problemscircumcirclecircles