MathDB

6

Part of 2009 Sharygin Geometry Olympiad

Problems(4)

Find the locus of excenters

Source: I.F.Sharygin contest 2009 - Correspondence round - Problem 6

5/31/2009
Find the locus of excenters of right triangles with given hypotenuse.
geometrygeometry proposed
4 equal polygons with no common interior but common boundary segment

Source: 2009 Sharygin Geometry Olympiad Final Round problem 6 grade 8

7/26/2018
Can four equal polygons be placed on the plane in such a way that any two of them don't have common interior points, but have a common boundary segment?
(S.Markelov)
geometrypolygon
let triangle with AB - BC =AC / \sqrt2

Source: 2009 Sharygin Geometry Olympiad Final Round problem 6 grade 9

7/26/2018
Given triangle ABCABC such that AB−BC=AC2AB- BC = \frac{AC}{\sqrt2} . Let MM be the midpoint of ACAC, and NN be the foot of the angle bisector from BB. Prove that ∠BMC+∠BNC=90o\angle BMC + \angle BNC = 90^o.
(A.Akopjan)
geometryangles
CGI is right iff GM // AB

Source: 2009 Sharygin Geometry Olympiad Final Round problem 6 grade 10

7/26/2018
Let M,IM, I be the centroid and the incenter of triangle ABC,A1ABC, A_1 and B1B_1 be the touching points of the incircle with sides BCBC and AC,GAC, G be the common point of lines AA1AA_1 and BB1BB_1. Prove that angle ∠CGI\angle CGI is right if and only if GM//ABGM // AB.
(A.Zaslavsky)
geometryCentroidincenterparallel