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 2 of 4: Use the four tangency points

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OM\perp AB,\quad ON\perp BC,\quad OP\perp CD,\quad OQ\perp DA,\qquad MP=NQ=a

Detailed analysis

Draw the two excircles centered at $O$ . Let them touch $AB,BC,CD,DA$ at $M,N,P,Q$ respectively. Because opposite sides of the square are parallel, $M,O,P$ are collinear and $N,O,Q$ are collinear. The distance between the parallel sides is the side length, hence $MP=NQ=a$ .