MathDB

2

Part of 2009 Sharygin Geometry Olympiad

Problems(4)

Segments bisect triangle's perimeter

Source: I.F.Sharygin contest 2009 - Correspondence round - Problem 2

5/31/2009
Given nonisosceles triangle ABC ABC. Consider three segments passing through different vertices of this triangle and bisecting its perimeter. Are the lengths of these segments certainly different?
geometryperimeterinequalitiesgeometry proposed
cyclic quadrilateral divided into 4 quadrilaterals , cyclic with equal r

Source: 2009 Sharygin Geometry Olympiad Final Round problem 2 grade 8

7/26/2018
A cyclic quadrilateral is divided into four quadrilaterals by two lines passing through its inner point. Three of these quadrilaterals are cyclic with equal circumradii. Prove that the fourth part also is cyclic quadrilateral and its circumradius is the same.
(A.Blinkov)
geometrycyclic quadrilateralcircumradius
inequality with circumradii of 4 triangles of convex ABCD

Source: 2009 Sharygin Geometry Olympiad Final Round problem 2 grade 9

7/26/2018
Given a convex quadrilateral ABCDABCD. Let Ra,Rb,RcR_a, R_b, R_c and RdR_d be the circumradii of triangles DAB,ABC,BCD,CDADAB, ABC, BCD, CDA. Prove that inequality Ra<Rb<Rc<RdR_a < R_b < R_c < R_d is equivalent to 180o−∠CDB<∠CAB<∠CDB180^o - \angle CDB < \angle CAB < \angle CDB .
(O.Musin)
inequalitiesconvex quadrilateralcircumradius
if intersections of sidelines &amp; diagonals pass through centroid =&gt; trapezoid

Source: 2009 Sharygin Geometry Olympiad Final Round problem 2 grade 10

7/26/2018
Given quadrilateral ABCDABCD. Its sidelinesAB AB and CDCD intersect in point KK. It's diagonals intersect in point LL. It is known that line KLKL pass through the centroid of ABCDABCD. Prove that ABCDABCD is trapezoid.
(F.Nilov)
geometrytrapezoid