3

Part of 2011 Oral Moscow Geometry Olympiad

Problems(2)

dividing a diagonal of a 2x2 square into 6 equal parts with unmarked ruler

Source: 2011 Oral Moscow Geometry Olympiad grades 8-9 p3

2\times 2

square was cut from a squared sheet of paper. Using only a ruler without divisions and without going beyond the square, divide the diagonal of the square into

6

Rulergeometryconstructiondiagonal

concurrency related to a trapezoid and a circle

Source: 2011 Oral Moscow Geometry Olympiad grades 10-11 p3

A non-isosceles trapezoid

ABCD

AB // CD

) is given. An arbitrary circle passing through points

A

B

intersects the sides of the trapezoid at points

P

Q

, and the intersect the diagonals at points

M

N

. Prove that the lines

PQ, MN

CD

are concurrent.

geometrytrapezoidconcurrencyconcurrentcircle