3

Part of 2016 Oral Moscow Geometry Olympiad

Problems(2)

distance point from a line is half the hypotenuse

Source: 2016 Oral Moscow Geometry Olympiad grades 8-9 p3

A circle with center

O

passes through the ends of the hypotenuse of a right-angled triangle and intersects its legs at points

M

K

. Prove that the distance from point

O

MK

is half the hypotenuse.

right triangledistancegeometry

equal areas by two squares inside another square

Source: 2016 Oral Moscow Geometry Olympiad grades 10-11 p3

Two squares are arranged as shown in the picture. Prove that the areas of shaded quadrilaterals are equal. https://3.bp.blogspot.com/-W50DOuizFvY/XT6wh3-L6sI/AAAAAAAAKaw/pIW2RKmttrwPAbrKK3bpahJz7hfIZwM8QCK4BGAYYCw/s400/Oral%2BSharygin%2B2016%2B10.11%2Bp3.png

geometrySquaressquareareasequal area