MathDB

5

Part of 2014 Oral Moscow Geometry Olympiad

Problems(2)

perimeter of the triangle DEF is two times larger than the side BC

Source: 2014 Oral Moscow Geometry Olympiad grades 8-9 p5

8/9/2019
Segment ADAD is the diameter of the circumscribed circle of an acute-angled triangle ABCABC. Through the intersection of the altitudes of this triangle, a straight line was drawn parallel to the side BCBC, which intersects sides ABAB and ACAC at points EE and FF, respectively. Prove that the perimeter of the triangle DEFDEF is two times larger than the side BCBC.
circumcirclegeometryperimeter
P in circumcircle of an equilateral ABC, AP cuts BC at A', area of A'B'C'

Source: 2014 Oral Moscow Geometry Olympiad grades 10-11 p5

8/9/2019
Given a regular triangle ABCABC, whose area is 11, and the point PP on its circumscribed circle. Lines AP,BP,CPAP, BP, CP intersect, respectively, lines BC,CA,ABBC, CA, AB at points A′,B′,C′A', B', C'. Find the area of the triangle A′B′C′A'B'C'.
geometrycircumcirclearea of a triangleareaEquilateralEquilateral Triangle