MathDB

4

Part of 2008 Sharygin Geometry Olympiad

Problems(4)

Triangle and two angles

Source: Sharygin contest. The final raund. 2008. Grade 8. First day. Problem 4

8/31/2008
(F.Nilov, A.Zaslavsky) Let CC0 CC_0 be a median of triangle ABC ABC; the perpendicular bisectors to AC AC and BC BC intersect CC0 CC_0 in points A′ A', B′ B'; C1 C_1 is the meet of lines AA′ AA' and BB′ BB'. Prove that \angle C_1CA \equal{} \angle C_0CB.
symmetrygeometryparallelogramangle bisectorgeometry unsolved
Circle passes through the circumcenter

Source: Sharygin contest. The final raund. 2008. Grade 9. First day. Problem 4

8/31/2008
(F.Nilov, A.Zaslavsky) Let CC0 CC_0 be a median of triangle ABC ABC; the perpendicular bisectors to AC AC and BC BC intersect CC0 CC_0 in points Ac A_c, Bc B_c; C1 C_1 is the common point of AAc AA_c and BBc BB_c. Points A1 A_1, B1 B_1 are defined similarly. Prove that circle A1B1C1 A_1B_1C_1 passes through the circumcenter of triangle ABC ABC.
geometrycircumcircleangle bisectorgeometry unsolved
Locus of incenters of triangles

Source: Sharygin contest. The final raund. 2008. Grade 10. First day. Problem 4

8/31/2008
(A.Zaslavsky) Given three points C0 C_0, C1 C_1, C2 C_2 on the line l l. Find the locus of incenters of triangles ABC ABC such that points A A, B B lie on l l and the feet of the median, the bisector and the altitude from C C coincide with C0 C_0, C1 C_1, C2 C_2.
geometryincentergeometry unsolved
Sum of squares of some two sides

Source: Sharygin contest 2008. The correspondence round. Problem 4

9/3/2008
(D.Shnol, 8--9) The bisectors of two angles in a cyclic quadrilateral are parallel. Prove that the sum of squares of some two sides in the quadrilateral equals the sum of squares of two remaining sides.
geometrycyclic quadrilateralgeometry proposed