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4x^3 = (a^2+b^2 + c^2)x + abc, computational geometry, max area of DECB

Source: 2002 Romania JBMO TST5 p3

May 16, 2020

geometrymaxareamidpointsidelengths

Problem Statement

Let

ABC

be a triangle and

a = BC, b = CA

and

c = AB

be the lengths of its sides. Points

D

and

E

lie in the same halfplane determined by

BC

A

. Suppose that

DB = c, CE = b

and that the area of

DECB

is maximal. Let

F

be the midpoint of

DE

and let

FB = x

. Prove that

FC = x

and

4x^3 = (a^2+b^2 + c^2)x + abc