MathDB

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Part of 2019 Sharygin Geometry Olympiad

Problems(4)

Sharygin CR 2019 P1

Source:

3/6/2019
Let AA1AA_1, CC1CC_1 be the altitudes of ΔABC\Delta ABC, and PP be an arbitrary point of side BCBC. Point QQ on the line ABAB is such that QP=PC1QP = PC_1, and point RR on the line ACAC is such that RP=CPRP = CP. Prove that QA1RAQA_1RA is a cyclic quadrilateral.
geometry
Midpoint on circumcircle

Source: Sharygin 2019 finals Day 1 Grade 8 P1

7/30/2019
A trapezoid with bases ABAB and CDCD is inscribed into a circle centered at OO. Let APAP and AQAQ be the tangents from AA to the circumcircle of triangle CDOCDO. Prove that the circumcircle of triangle APQAPQ passes through the midpoint of ABAB.
geometrySharygin Geometry Olympiadcircumcircle
Undescribed triangle?

Source: Sharygin 2019 Finals Day 1 Grade 9 P1

7/30/2019
A triangle OABOAB with ∠A=90∘\angle A=90^{\circ} lies inside another triangle with vertex OO. The altitude of OABOAB from AA until it meets the side of angle OO at MM. The distances from MM and BB to the second side of angle OO are 22 and 11 respectively. Find the length of OAOA.
geometrySharygin Geometry Olympiad
EF bisects A'K in triangle with 45° angle

Source: Sharygin 2019 Finals Day 1 Grade 10 P1

7/30/2019
Given a triangle ABCABC with ∠A=45∘\angle A = 45^\circ. Let A′A' be the antipode of AA in the circumcircle of ABCABC. Points EE and FF on segments ABAB and ACAC respectively are such that A′B=BEA'B = BE, A′C=CFA'C = CF. Let KK be the second intersection of circumcircles of triangles AEFAEF and ABCABC. Prove that EFEF bisects A′KA'K.
geometry