MathDB

5

Part of 2019 Oral Moscow Geometry Olympiad

Problems(2)

locus of intersections of perpendicular bisectors of two equal segments is line

Source: 2019 Oral Moscow Geometry Olympiad grades 8-9 p5

5/21/2019
Given the segment PQ PQ and a circle . A chord ABAB moves around the circle, equal to PQPQ. Let TT be the intersection point of the perpendicular bisectors of the segments APAP and BQBQ. Prove that all points of TT thus obtained lie on one line.
geometryperpendicular bisectorLocusLocus problemscollinear
two lines concurrent with a circumcircle

Source: 2019 Oral Moscow Geometry Olympiad grades 10-11 p5

5/22/2019
On sides ABAB and BCBC of a non-isosceles triangle ABCABC are selected points C1C_1 and A1A_1 such that the quadrilateral AC1A1CAC_1A_1C is cyclic. Lines CC1CC_1 and AA1AA_1 intersect at point PP. Line BPBP intersects the circumscribed circle of triangle ABCABC at the point QQ. Prove that the lines QC1QC_1 and CMCM, where MM is the midpoint of A1C1A_1C_1, intersect at the circumscribed circles of triangle ABCABC.
geometrycircumcirclecyclic quadrilateralconcurrent