MathDB

5

Part of 2017 Sharygin Geometry Olympiad

Problems(3)

Inner tangent passes through midpoint

Source: Sharygin Finals 2017, Problem 8.5

8/4/2017
A square ABCDABCD is given. Two circles are inscribed into angles AA and BB, and the sum of their diameters is equal to the sidelength of the square. Prove that one of their common tangents passes through the midpoint of ABAB.
geometryTangentsmidpointsSquares
Line parallel to base

Source: Sharygin Finals 2017, Problem 9.5

8/3/2017
Let BHb,CHcBH_b, CH_c be altitudes of an acute-angled triangle ABCABC. The line HbHcH_bH_c meets the circumcircle of ABCABC at points XX and YY. Points P,QP,Q are the reflections of X,YX,Y about AB,ACAB,AC respectively. Prove that PQ∥BCPQ \parallel BC.
Proposed by Pavel Kozhevnikov
geometrySharygin Geometry Olympiadparallelaltitudesreflection
Concyclic points and miquel point

Source: Sharygin 2017 Day 2 Problem 10.5 Grade 10

8/2/2017
10.5 Let BB′BB', CC′CC' be the altitudes of an acute triangle ABCABC. Two circles through AA and C′C' are tangent to BCBC at points PP and QQ. Prove that A,B′,P,QA, B', P, Q are concyclic.
geometry