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 2 of 5: Express the four side fragments by similarity

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AM=NC=-\frac{s}{h}AB,\qquad CQ=PA=\frac{t}{h}AB

Detailed analysis

Along each side of the equal-sided quadrilateral, the coordinate on a line perpendicular to $BD$ changes proportionally. At position $s$ , the two fragments from $A$ and $C$ have length $-sAB/h$ ; at position $t$ , the fragments $PA$ and $CQ$ have length $tAB/h$ . Thus $AM=NC=-sAB/h$ and $CQ=PA=tAB/h$ .