Problem 2

Suppose

ABCD

is a square of side length

a

. Two parallel lines

\ell_1

and

\ell_2

in the plane are

a

units apart. The square is placed so that

AB

and

AD

meet

\ell_1

E

and

F

, while

CB

and

CD

meet

\ell_2

G

and

H

. If the perimeters of

\triangle AEF

and

\triangle CGH

are

m_1

and

m_2

, prove that

m_1+m_2

is constant, regardless of the placement.

Step 1 of 4: Locate the common excentre

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O=EH\cap FG

Detailed analysis

Let $O=EH\cap FG$ . The distances from $G$ to the lines $FD$ and $EF$ are both $a$ , so $FG$ bisects the relevant angle at $F$ ; similarly $EH$ bisects the relevant angle at $E$ . Thus $O$ is an excentre of $\triangle AEF$ . The same argument at $G,H$ shows that it is also an excentre of $\triangle CGH$ .