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 3 of 4: Express the bases by equal tangent lengths

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EF=EM+FQ,\qquad GH=GN+HP

Detailed analysis

From equal tangent lengths from each external point to an excircle, $EF=EM+FQ$ and $GH=GN+HP$ . Therefore $m_1=AE+AF+EF=(AM+AQ)+(EM+FQ)=OM+OQ$ , while $m_2=CG+CH+GH=ON+OP$ .