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Problem 1

Let ABCDABCD be a quadrilateral with AB=BC=CD=DAAB=BC=CD=DA. Let MNMN and PQPQ be segments perpendicular to diagonal BDBD, with M∈ADM\in AD, N∈DCN\in DC, P∈ABP\in AB, and Q∈BCQ\in BC, and with distance d>BD/2d>BD/2 between them. Show that the perimeter of hexagon AMNCQPAMNCQP does not depend on the positions of MNMN and PQPQ while their distance remains dd.
Step 4 of 5: Sum the six perimeter pieces
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L=AM+MN+NC+CQ+QP+PA=2AC+2AB−ACh(t−s)L=AM+MN+NC+CQ+QP+PA=2AC+\frac{2AB-AC}{h}(t-s)
Detailed analysis

Substitute the four side fragments and two cross-sections: L=(−sAB/h)+AC(1+s/h)+(−sAB/h)+(tAB/h)+AC(1−t/h)+(tAB/h)L=(-sAB/h)+AC(1+s/h)+(-sAB/h)+(tAB/h)+AC(1-t/h)+(tAB/h). Collecting terms gives L=2AC+((2AB−AC)/h)(t−s)L=2AC+((2AB-AC)/h)(t-s).