MathDB

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Part of 2014 Sharygin Geometry Olympiad

Problems(4)

Restoring a Point in a Right Triangle

Source: Sharygin Geometry Olympiad 2014 - Problem 1

11/15/2014
A right-angled triangle ABCABC is given. Its catheus ABAB is the base of a regular triangle ADBADB lying in the exterior of ABCABC, and its hypotenuse ACAC is the base of a regular triangle AECAEC lying in the interior of ABCABC. Lines DEDE and ABAB meet at point MM. The whole configuration except points AA and BB was erased. Restore the point MM.
geometry unsolvedgeometry
equal segments starting with the incircle of a right triangle

Source: 2014 Sharygin Geometry Olympiad Final Round 8.1

8/3/2018
The incircle of a right-angled triangle ABCABC touches its catheti ACAC and BCBC at points B1B_1 and A1A_1, the hypotenuse touches the incircle at point C1C_1. Lines C1A1C_1A_1 and C1B1C_1B_1 meet CACA and CBCB respectively at points B0B_0 and A0A_0. Prove that AB0=BA0AB_0 = BA_0.
(J. Zajtseva, D. Shvetsov )
geometryright triangle
cyclic ABCD: AC > BD iff (AD-BC)(AB-CD) > 0.

Source: 2014 Sharygin Geometry Olympiad Final Round 9.1

8/3/2018
Let ABCDABCD be a cyclic quadrilateral. Prove that AC>BDAC > BD if and only if (AD−BC)(AB−CD)>0(AD-BC)(AB- CD) > 0.
(V. Yasinsky)
geometrycyclic quadrilateralgeometric inequality
angle chasing with an isosceles inscribed in a square

Source: 2014 Sharygin Geometry Olympiad Final Round 10.1

8/3/2018
The vertices and the circumcenter of an isosceles triangle lie on four different sides of a square. Find the angles of this triangle.
(I. Bogdanov, B. Frenkin)
Angle Chasingisoscelesgeometrysquare