MathDB

3

Part of 2010 Sharygin Geometry Olympiad

Problems(4)

Prove that the quadrilateral XA'BC' is cyclic (3)

Source:

10/28/2010
Points A′,B′,C′A', B', C' lie on sides BC,CA,ABBC, CA, AB of triangle ABC.ABC. for a point XX one has ∠AXB=∠A′C′B′+∠ACB\angle AXB =\angle A'C'B' + \angle ACB and ∠BXC=∠B′A′C′+∠BAC.\angle BXC = \angle B'A'C' +\angle BAC. Prove that the quadrilateral XA′BC′XA'BC' is cyclic.
geometrycircumcirclegeometry proposed
equal segments starting with segment bisectors in a convex ABCD

Source: Sharygin 2010 Final 8.3

10/2/2018
Let ABCDABCD be a convex quadrilateral and KK be the common point of rays ABAB and DCDC. There exists a point PP on the bisectrix of angle AKDAKD such that lines BPBP and CPCP bisect segments ACAC and BDBD respectively. Prove that AB=CDAB = CD.
geometryequal segmentsbisectionangle bisector
equilateral triangles, 1 vertex in 1 line, concurrency wanted

Source: Sharygin 2010 Final 9.3

10/6/2018
Points X,Y,ZX,Y,Z lies on a line (in indicated order). Triangles XABXAB, YBCYBC, ZCDZCD are regular, the vertices of the first and the third triangle are oriented counterclockwise and the vertices of the second are opposite oriented. Prove that ACAC, BDBD and XYXY concur.
V.A.Yasinsky
geometryconcurrencyconcurrentEquilateral Triangle
convex n-gons with each side of 1st > each side of 2nd, what about diagonals?

Source: Sharygin 2010 Final 10.3

11/25/2018
All sides of a convex polygon were decreased in such a way that they formed a new convex polygon. Is it possible that all diagonals were increased?
geometryconvex polygonInequalitygeometric inequalitydiagonalsdiagonal