MathDB

6

Part of 2010 Oral Moscow Geometry Olympiad

Problems(2)

line touches a fixed circle., perpendicular bisectors and circumcircle related

Source: 2010 Oral Moscow Geometry Olympiad grades 8-9 p6

8/16/2020
Perpendicular bisectors of the sides BCBC and ACAC of an acute-angled triangle ABCABC intersect lines ACAC and BCBC at points MM and NN. Let point CC move along the circumscribed circle of triangle ABCABC, remaining in the same half-plane relative to ABAB (while points AA and BB are fixed). Prove that line MNMN touches a fixed circle.
geometryperpendicular bisectorcircumcirclefixedcircle
concurrency, lines passing through midpoint of altitudes // OA_i

Source: 2010 Oral Moscow Geometry Olympiad grades 10-11 p6

5/8/2020
In a triangle ABC,OABC, O is the center of the circumscribed circle. Line aa passes through the midpoint of the altitude of the triangle from the vertex AA and is parallel to OAOA. Similarly, the straight lines bb and cc are defined. Prove that these three lines intersect at one point.
geometryconcurrencyconcurrentCircumcenterparallel