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The 3-4-5 homothety bisection (BMT 2019 Geo #6)

Source:

May 12, 2019

geometrygeometric transformationhomothety

Problem Statement

Let

\triangle ABE

be a triangle with

\frac{AB}{3} = \frac{BE}{4} = \frac{EA}{5}

. Let

D \neq A

be on line

\overline{AE}

such that

AE = ED

and

D

is closer to

E

than to

A

. Moreover, let

C

be a point such that

BCDE

is a parallelogram. Furthermore, let

M

be on line

\overline{CD}

such that

\overline{AM}

bisects

\angle BAE

, and let

P

be the intersection of

\overline{AM}

and

\overline{BE}

. Compute the ratio of

PM

to the perimeter of

\triangle ABE

.

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