MathDB

2

Part of 2011 Sharygin Geometry Olympiad

Problems(4)

Projection of a vertex to an angle bisector

Source: 2011 Sharygin Geometry Olympiad #2

5/22/2014
Let ABCABC be a triangle with sides AB=4AB = 4 and AC=6AC = 6. Point HH is the projection of vertex BB to the bisector of angle AA. Find MHMH, where MM is the midpoint of BCBC.
geometry unsolvedgeometry
restore the original paper rectangle using compass and ruler

Source: Sharygin 2011 Final 8.2

12/13/2018
Peter made a paper rectangle, put it on an identical rectangle and pasted both rectangles along their perimeters. Then he cut the upper rectangle along one of its diagonals and along the perpendiculars to this diagonal from two remaining vertices. After this he turned back the obtained triangles in such a way that they, along with the lower rectangle form a new rectangle. Let this new rectangle be given. Restore the original rectangle using compass and ruler.
geometryrectangleconstructionpaper
anglechasing , circumcenter wanted

Source: Sharygin 2011 Final 9.2

12/16/2018
In triangle ABC,∠B=2∠CABC, \angle B = 2\angle C. Points PP and QQ on the medial perpendicular to CBCB are such that ∠CAP=∠PAQ=∠QAB=∠A3\angle CAP = \angle PAQ = \angle QAB = \frac{\angle A}{3} . Prove that QQ is the circumcenter of triangle CPBCPB.
geometrycircumcircleAngle Chasingperpendicular bisector
cyclic PQRS related to midpoints and touchpoints of incircle of ABCD

Source: Sharygin 2011 Final 10.2

3/31/2019
Quadrilateral ABCDABCD is circumscribed. Its incircle touches sides AB,BC,CD,DAAB, BC, CD, DA in points K,L,M,NK, L, M, N respectively. Points A′,B′,C′,D′A', B', C', D' are the midpoints of segments LM,MN,NK,KLLM, MN, NK, KL. Prove that the quadrilateral formed by lines AA′,BB′,CC′,DD′AA', BB', CC', DD' is cyclic.
geometrycyclic quadrilateraltangential quadrilateralmidpoints