Problem 1

Let

ABCD

be a quadrilateral with

AB=BC=CD=DA

. Let

MN

and

PQ

be segments perpendicular to diagonal

BD

, with

M\in AD

N\in DC

P\in AB

, and

Q\in BC

, and with distance

d>BD/2

between them. Show that the perimeter of hexagon

AMNCQP

does not depend on the positions of

MN

and

PQ

while their distance remains

d

Step 5 of 5: Conclude independence from the positions

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L=2AC+\frac{2AB-AC}{h}d

Detailed analysis

Because $t-s=d$ is fixed, the final expression $L=2AC+((2AB-AC)/h)d$ is constant. Therefore the perimeter of $AMNCQP$ does not depend on the individual positions of $MN$ and $PQ$ .