5

Part of 2021 Oral Moscow Geometry Olympiad

Problems(2)

geo inequality with inscribed trapezoid

Source: 2021 Oral Moscow Geometry Olympiad grades 8-9 p5

The trapezoid is inscribed in a circle. Prove that the sum of distances from any point of the circle to the midpoints of the lateral sides are not less than the diagonal of the trapezoid.

geometrycircumcirclegeometric inequalitytrapezoidisosceles

Two triangles have the same nine point centers

Source: 2021 Oral Moscow Geometry Olympiad grades 10-11 p5

ABC

I

O

be its incenter and circumcenter respectively.

A'

is symmetric to

O

with respect to line

AI

B'

C'

are defined similarly. Prove that the nine-point centers of triangles

ABC

A'B'C'

geometryincentercircumcircle