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 3 of 5: Express the two cut lengths

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MN=AC\left(1+\frac{s}{h}\right),\qquad PQ=AC\left(1-\frac{t}{h}\right)

Detailed analysis

The same similarity at the two ends of the diagonal gives the length of a perpendicular cross-section as a linear interpolation between the diagonal $AC$ at $O$ and zero at $D$ or $B$ . Hence $MN=AC(1+s/h)$ and $PQ=AC(1-t/h)$ .