MathDB

Let

ABCD

be a convex quadrilateral.

AC

and

BD

meet at

P

, with

\angle APD=60^{\circ}

. Let

E,F,G

, and

H

be the midpoints of

AB,BC,CD

and

DA

respectively. Find the greatest positive real number

k

for which

EG+3HF\ge kd+(1-k)s

where

s

is the semi-perimeter of the quadrilateral

ABCD

and

d

is the sum of the lengths of its diagonals. When does the equality hold?

Problem Statement