MathDB

7

Part of 2019 Sharygin Geometry Olympiad

Problems(3)

Sharygin CR 2019 P7

Source:

3/6/2019
Let AHAAH_A, BHBBH_B, CHCCH_C be the altitudes of the acute-angled ΔABC\Delta ABC. Let XX be an arbitrary point of segment CHCCH_C, and PP be the common point of circles with diameters HCXH_CX and BC, distinct from HCH_C. The lines CPCP and AHAAH_A meet at point QQ, and the lines XPXP and ABAB meet at point RR. Prove that A,P,Q,R,HBA, P, Q, R, H_B are concyclic.
geometry
Medial line of a triangle

Source: Sharygin 2019 Finals Day 2 Grade 9 P3

7/31/2019
Let the incircle ω\omega of △ABC\triangle ABC touch ACAC and ABAB at points EE and FF respectively. Points XX , YY of ω\omega are such that ∠BXC=∠BYC=90∘\angle BXC=\angle BYC=90^{\circ} . Prove that EFEF and XYXY meet on the medial line of ABCABC .
locus, incenter, incircle, parallel related

Source: Sharygin 2019 10 p7

8/15/2019
Let PP be an arbitrary point on side BCBC of triangle ABCABC. Let KK be the incenter of triangle PABPAB. Let the incircle of triangle PACPAC touch BCBC at FF. Point GG on CKCK is such that FG//PKFG // PK. Find the locus of GG.
geometryincenterLocusincircle