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6

Part of 2017 Oral Moscow Geometry Olympiad

Problems(2)

another locus, center of the circumcircle moves on a straight line

Source: 2017 Oral Moscow Geometry Olympiad grades 8-9 p6

7/26/2019
Around triangle ABCABC with acute angle C is circumscribed a circle. On the arc ABAB, which does not contain point CC, point DD is chosen. Point D′D' is symmetric on point DD with respect to line ABAB. Straight lines AD′AD' and BD′BD' intersect segments BCBC and ACAC at points EE and FF. Let point CC move along its arc ABAB. Prove that the center of the circumscribed circle of a triangle CEFCEF moves on a straight line.
geometrycircumcircleCircumcenterLocus
Geometry

Source: Sharygin Oral Geometry Olympiad 2017.Senior League(10-11) P6

4/16/2017
Given acute angled traingle ABCABC and altitudes AA1AA_1, BB1BB_1, CC1CC_1. Let MM midpoint of BCBC. PP point of intersection of circles (AB1C1)(AB_1C_1) and (ABC)(ABC) . TT is point of intersection of tangents to (ABC)(ABC) at BB and CC. SS point of intersection of ATAT and (ABC)(ABC). Prove that P,A1,SP,A_1,S and midpoint of MTMT collinear.
geometrycircumcircle