MathDB

5

Part of 2009 Sharygin Geometry Olympiad

Problems(4)

Determine angle A

Source: I.F.Sharygin contest 2009 - Correspondence round - Problem 5

5/31/2009
Given triangle ABC ABC. Point O O is the center of the excircle touching the side BC BC. Point O1 O_1 is the reflection of O O in BC BC. Determine angle A A if O1 O_1 lies on the circumcircle of ABC ABC.
geometrygeometric transformationreflectioncircumcirclegeometry proposed
angle: projection of B on <C-bisector, touchpoint incircle with BC, B

Source: 2009 Sharygin Geometry Olympiad Final Round problem 5 grade 8

7/26/2018
Given triangle ABCABC. Point MM is the projection of vertex BB to bisector of angle CC. KK is the touching point of the incircle with side BCBC. Find angle ∠MKB\angle MKB if ∠BAC=α\angle BAC = \alpha
(V.Protasov)
geometryangleprojection
half of triangles formed by n points on a circle are acute

Source: 2009 Sharygin Geometry Olympiad Final Round problem 5 grade 9

7/26/2018
Let nn points lie on the circle. Exactly half of triangles formed by these points are acute-angled. Find all possible nn.
(B.Frenkin)
geometrycombinatorial geometrypointsacute triangle
rhombus incribed in triangle, 3 circumenters and an intersecion collinear

Source: 2009 Sharygin Geometry Olympiad Final Round problem 5 grade 10

7/26/2018
Rhombus CKLNCKLN is inscribed into triangle ABCABC in such way that point LL lies on side ABAB, point NN lies on side ACAC, point KK lies on side BCBC. O1,O2O_1, O_2 and OO are the circumcenters of triangles ACL,BCLACL, BCL and ABCABC respectively. Let PP be the common point of circles ANLANL and BKLBKL, distinct from LL. Prove that points O1,O2,OO_1, O_2, O and PP are concyclic.
(D.Prokopenko)
geometryrhombuscollinearCircumcenter