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AE/BE =AC xCD / (AC^2 - CD^2) , right trapezoid with perp. diagonals

Source: 2012 Grand Duchy of Lithuania, Mathematical Contest p2 (Baltic Way TST)

October 2, 2020

geometrytrapezoidright angle

Problem Statement

The base

AB

of a trapezium

ABCD

is longer than the base

CD

, and

\angle ADC

is a right angle. The diagonals

AC

and

BD

are perpendicular. Let

E

be the foot of the altitude from

D

to the line

BC

. Prove that

\frac{AE}{BE} =\frac{ AC \cdot CD}{AC^2 - CD^2}