MathDB

3

Part of 2017 Sharygin Geometry Olympiad

Problems(3)

Concentric circles in centroid configuration

Source: Sharygin Finals 2017, Problem 8.3

8/4/2017
Let AD,BEAD, BE and CFCF be the medians of triangle ABCABC. The points XX and YY are the reflections of FF about ADAD and BEBE, respectively. Prove that the circumcircles of triangles BEXBEX and ADYADY are concentric.
geometryCentroidconcentricgeometric transformationreflection
Obtuse angle at variable midpoint

Source: Sharygin Finals 2017, Problem 9.3

8/3/2017
The angles BB and CC of an acute-angled​ triangle ABCABC are greater than 60∘60^\circ. Points P,QP,Q are chosen on the sides AB,ACAB,AC respectively so that the points A,P,QA,P,Q are concyclic with the orthocenter HH of the triangle ABCABC. Point KK is the midpoint of PQPQ. Prove that ∠BKC>90∘\angle BKC > 90^\circ.
Proposed by A. Mudgal
geometric inequalitygeometry
Power×area

Source: Sharygin final round 2017

7/31/2017
ABCDABCD is convex quadrilateral. If WaW_a is product of power of AA about circle BCDBCD and area of triangle BCDBCD. And define Wb,Wc,WdW_b,W_c,W_d similarly.prove Wa+Wb+Wc+Wd=0W_a+W_b+W_c+W_d=0
geometry