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 1 of 5: Parameterize the two cuts by signed positions

Previous step Next step

h=\frac{BD}{2},\qquad s<0<t,\qquad t-s=d

Detailed analysis

Let $O$ be the midpoint of $BD$ and put $h=BD/2$ . Measure signed distance along $BD$ from $O$ . Since $MN$ meets $AD,DC$ and $PQ$ meets $AB,BC$ , write their positions as $s<0$ and $t>0$ . Their perpendicular separation is $t-s=d$ .