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Lots of perpendiculars

Source: Baltic Way 2017 Problem 14

November 11, 2017

geometry

Problem Statement

Let

P

be a point inside the acute angle

\angle BAC

. Suppose that

\angle ABP = \angle ACP = 90^{\circ}

. The points

D

and

F

are on the segments

G

and

CA

, respectively, such that

BD = BP

and

CP = CE

. The points

F

and

G

are on the segments

AC

and

AB

, respectively, such that

DF

is perpendicular to

AB

and

EG

is perpendicular to

AC

. Show that

PF = PG

.

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