MathDB

2

Part of 2017 Sharygin Geometry Olympiad

Problems(3)

Angle bisector through circumcenter

Source: Sharygin Finals 2017, Problem 8.2

8/4/2017
Let HH and OO be the orthocenter and circumcenter of an acute-angled triangle ABCABC, respectively. The perpendicular bisector of BHBH meets ABAB and BCBC at points A1A_1 and C1C_1, respectively. Prove that OBOB bisects the angle A1OC1A_1OC_1.
geometrycircumcircleperpendicular bisectorangle bisector
Mixtilinear touchpoint is midpoint of side

Source: Sharygin Finals 2017, Problem 9.2

8/3/2017
Let II be the incenter of a triangle ABCABC, MM be the midpoint of ACAC, and WW be the midpoint of arc ABAB of the circumcircle not containing CC. It is known that ∠AIM=90∘\angle AIM = 90^\circ. Find the ratio CI:IWCI:IW.
geometryratio
Easy inequality in ABC

Source: Sharygin final round 2017

7/31/2017
If ABCABC is acute triangle, prove distance from each vertex to corresponding excentre is less than sum of two greatest side of triangle
inequalities proposedgeometry