MathDB

Let

ABC

be an acute-angled triangle such that

AB+BC=4AC

. Let

D

AC

such that

BD

is angle bisector of

\angle ABC

. In the segment

BD

, points

P

and

Q

are marked such that

BP=2DQ

. The perpendicular line to

BD

, passing by

Q

, cuts the segments

AB

and

BC

X

and

Y

, respectively. Let

L

be the parallel line to

AC

passing by

P

. The point

B

is in a different half-plane(with respect to the line

L

) of the points

X

and

Y

. An ant starts a run in the point

X

, goes to a point in the line

AC

, after that goes to a point in the line

L

, returns to a point in the line

AC

and finishes in the point

Y

. Prove that the least length of the ant's run is equal to

4XY

Problem Statement