MathDB

6

Part of 2007 Oral Moscow Geometry Olympiad

Problems(2)

fixed circle, all segments XY are tangent to same circle, projections

Source: 2007 Oral Moscow Geometry Olympiad grades 8-9 p6

10/18/2020
A point PP is fixed inside the circle. CC is an arbitrary point of the circle, ABAB is a chord passing through point BB and perpendicular to the segment BCBC. Points XX and YY are projections of point BB onto lines ACAC and BCBC. Prove that all line segments XYXY are tangent to the same circle.
(A. Zaslavsky)
geometryfixedtangent
fixed point wanted, intersection of tangents to circle, perpendicular rays

Source: 2007 Oral Moscow Geometry Olympiad grades 10-11 p6

10/19/2020
A circle and a point PP inside it are given. Two arbitrary perpendicular rays starting at point PP intersect the circle at points AA and BB. Point XX is the projection of point PP onto line AB,YAB, Y is the intersection point of tangents to the circle drawn through points AA and BB. Prove that all lines XYXY pass through the same point.
(A. Zaslavsky)
geometryfixedTangentsFixed point