MathDB

strange area condition given (GCE)=1/2 (a^3/b+ab) inside a rectangle

Source: JBMO Shortlist 2018 G5

July 22, 2019

geometryrectanglearea of a trianglesimilar trianglesperpendicular

Problem Statement

Given a rectangle

ABCD

such that

AB = b > 2a = BC

, let

E

be the midpoint of

AD

. On a line parallel to

AB

through point

E

, a point

G

is chosen such that the area of

GCE

is

(GCE)= \frac12 \left(\frac{a^3}{b}+ab\right)

Point

H

is the foot of the perpendicular from

E

to

GD

and a point

I

is taken on the diagonal

AC

such that the triangles

ACE

and

AEI

are similar. The lines

BH

and

IE

intersect at

K

and the lines

CA

and

EH

intersect at

J

. Prove that

KJ \perp AB

.

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