MathDB

3

Part of 2018 Oral Moscow Geometry Olympiad

Problems(2)

circumcircle, excircle, equal segments, parallel wanted

Source: 2018 Oral Moscow Geometry Olympiad grades 8-9 p3

7/25/2019
On the extensions of sides CACA and ABAB of triangle ABCABC beyond points AA and BB, respectively, the segments AE=BCAE = BC and BF=ACBF = AC are drawn. A circle is tangent to segment BFBF at point NN, side BCBC and the extension of side ACAC beyond point CC. Point MM is the midpoint of segment EFEF. Prove that the line MNMN is parallel to the bisector of angle AA.
geometrycircumcircleexcircleparallel
orthocenters lie on a fixed circle, starting with another fixed circle

Source: 2018 Oral Moscow Geometry Olympiad grades 10-11 p3

7/25/2019
A circle is fixed, point AA is on it and point KK outside the circle. The secant passing through KK intersects circle at points PP and QQ. Prove that the orthocenters of the triangle APQAPQ lie on a fixed circle.
orthocenterfixedcirclegeometry