MathDB

5

Part of 2005 Oral Moscow Geometry Olympiad

Problems(2)

point construction, intersections of cevians with circumcircle, equilateral

Source: 2005 Oral Moscow Geometry Olympiad grades 8-9 p5

10/16/2020
The triangle ABCABC is inscribed in the circle. Construct a point PP such that the points of intersection of lines AP,BPAP, BP and CPCP with this circle are the vertices of an equilateral triangle.
(A. Zaslavsky)
geometrycircumcircleEquilateralconstruction
MA + MB + MC <=max (AB + BC, BC + AC, AC + AB) for interior M in ABC

Source: 2005 Oral Moscow Geometry Olympiad grades 10-11 p5

10/21/2020
An arbitrary point MM is chosen inside the triangle ABCABC. Prove that MA+MB+MC≤max(AB+BC,BC+AC,AC+AB)MA + MB + MC \le max (AB + BC, BC + AC, AC + AB).
(N. Sedrakyan)
geometrygeometric inequality