MathDB

7

Part of 2013 Sharygin Geometry Olympiad

Problems(4)

Prove that angle DBQ is right

Source: Sharygin First Round 2013, Problem 7

4/8/2013
Let BDBD be a bisector of triangle ABCABC. Points IaI_a, IcI_c are the incenters of triangles ABDABD, CBDCBD respectively. The line IaIcI_aI_c meets ACAC in point QQ. Prove that ∠DBQ=90∘\angle DBQ = 90^\circ.
geometryincenterAsymptoteprojective geometrygeometry proposed
4 points are centers of circles, 3 pairwise externally tangent, 3 tangent to 4th

Source: Sharygin 2013 Final 8.7

8/16/2018
In the plane, four points are marked. It is known that these points are the centers of four circles, three of which are pairwise externally tangent, and all these three are internally tangent to the fourth one. It turns out, however, that it is impossible to determine which of the marked points is the center of the fourth (the largest) circle. Prove that these four points are the vertices of a rectangle.
geometrycirclesrectangle
prove that all the points X are collinear as radius R changes, 3 circles related

Source: Sharygin 2013 Final 9.7

8/19/2018
Two fixed circles ω1\omega_1 and ω2\omega_2 pass through point OO. A circle of an arbitrary radius RR centered at OO meets ω1\omega_1 at points AA and BB, and meets ω2\omega_2 at points CC and DD. Let XX be the common point of lines ACAC and BDBD. Prove that all the points X are collinear as RR changes.
geometryLocuscollinearcircles
Solid geometry with unusual statement

Source: 10.7 Final Round of Sharygin geometry Olympiad 2013

8/8/2013
Given five fixed points in the space. It is known that these points are centers of five spheres, four of which are pairwise externally tangent, and all these point are internally tangent to the fifth one. It turns out that it is impossible to determine which of the marked points is the center of the largest sphere. Find the ratio of the greatest and the smallest radii of the spheres.
geometry3D geometrysphereratiogeometry proposed