MathDB

8

Part of 2012 Sharygin Geometry Olympiad

Problems(4)

Incircle and median

Source: Sharygin Geometry Olympiad 2012 - Problem 8

4/28/2012
Let BMBM be the median of right-angled triangle ABC(∠B=90∘)ABC (\angle B = 90^{\circ}). The incircle of triangle ABMABM touches sides AB,AMAB, AM in points A1,A2A_{1},A_{2}; points C1,C2C_{1}, C_{2} are defined similarly. Prove that lines A1A2A_{1}A_{2} and C1C2C_{1}C_{2} meet on the bisector of angle ABCABC.
geometryincentertrigonometrygeometric transformationhomothety
dividing square in different no of sides convex polygons, at least 1 is triangle

Source: 2012 Sharygin Geometry Olympiad Final Round 8.8

8/3/2018
A square is divided into several (greater than one) convex polygons with mutually different numbers of sides. Prove that one of these polygons is a triangle.
(A.Zaslavsky)
geometryconvex polygonsquare
prove that H is the incenter of triangle PQT

Source: 2012 Sharygin Geometry Olympiad Final Round 9.8

8/3/2018
Let AHAH be an altitude of an acute-angled triangle ABCABC. Points KK and LL are the projections of HH onto sides ABAB and ACAC. The circumcircle of ABCABC meets line KLKL at points PP and QQ, and meets line AHAH at points AA and TT. Prove that HH is the incenter of triangle PQTPQT.
(M.Plotnikov)
geometryincenter
collinear orthocenters

Source:

8/20/2012
A point MM lies on the side BCBC of square ABCDABCD. Let XX, YY , and ZZ be the incenters of triangles ABMABM, CMDCMD, and AMDAMD respectively. Let HxH_x, HyH_y, and HzH_z be the orthocenters of triangles AXBAXB, CYDCY D, and AZDAZD. Prove that HxH_x, HyH_y, and HzH_z are collinear.
geometryincentertrigonometryrhombusgeometry proposed