MathDB

Convex quadrilateral

ABCD

is given. Lines

BC

and

AD

intersect at point

O

, with

B

lying on the segment

OC

, and

A

on the segment

OD

I

is the center of the circle inscribed in the

OAB

triangle,

J

is the center of the circle exscribed in the triangle

OCD

touching the side of

CD

and the extensions of the other two sides. The perpendicular from the midpoint of the segment

IJ

on the lines

BC

and

AD

intersect the corresponding sides of the quadrilateral (not the extension) at points

X

and

Y

. Prove that the segment

XY

divides the perimeter of the quadrilateral

ABCD

in half, and from all segments with this property and ends on

BC

and

AD

, segment

XY

has the smallest length.

Problem Statement