MathDB

2

Part of 2010 Sharygin Geometry Olympiad

Problems(4)

Prove that OI is perpendicular to AB (2)

Source:

10/28/2010
Bisectors AA1AA_1 and BB1BB_1 of a right triangle ABC (∠C=90∘)ABC \ (\angle C=90^\circ ) meet at a point I.I. Let OO be the circumcenter of triangle CA1B1.CA_1B_1. Prove that OI⊥AB.OI \perp AB.
geometrycircumcircletrigonometryGaussEulergeometric transformationhomothety
locus of points C such points A,B,C can be covered by circle of radius 1

Source: Sharygin 2010 Final 8.2

10/2/2018
Two points AA and BB are given. Find the locus of points CC such that triangle ABCABC can be covered by a circle with radius 11.
(Arseny Akopyan)
geometryLocuscircle
intersecting triangles, at least 1 vertex is inside the circumcircle of other

Source: Sharygin 2010 Final 9.2

10/6/2018
Two intersecting triangles are given. Prove that at least one of their vertices lies inside the circumcircle of the other triangle.
(Here, the triangle is considered the part of the plane bounded by a closed three-part broken line, a point lying on a circle is considered to be lying inside it.)
geometrycircumcircleVertices
trig relation with two equal circles, each passing through center of other

Source: Sharygin 2010 Final 10.2

11/25/2018
Each of two equal circles ω1\omega_1 and ω2\omega_2 passes through the center of the other one. Triangle ABCABC is inscribed into ω1\omega_1, and lines AC,BCAC, BC touch ω2\omega_2 . Prove that cosA+cosB=1cosA + cosB = 1.
geometrytrigonometrycirclescircumcircletrigonometric