Problem 3

The cube

ABCDA'B'C'D'

has upper face

ABCD

and lower face

A'B'C'D'

, with

A

directly above

A'

and so on. Point

X

moves at constant speed around

ABCD

, and point

Y

moves at the same speed around

B'C'CB

. They leave

A

toward

B

and

B'

toward

C'

simultaneously. Find the locus of the midpoint of

XY

Step 1 of 6: Set coordinates and identify the relevant vertices

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A=(0,0,0),\quad C'=(1,1,1),\quad B=(1,0,0),\quad C=(1,1,0),\quad B'=(1,0,1)

A cube showing the two square perimeters followed by X and Y.

Detailed analysis

Use the unit cube with $A=(0,0,0)$ and $C'=(1,1,1)$ . Then $B=(1,0,0)$ , $C=(1,1,0)$ , and $B'=(1,0,1)$ , matching the two perimeters and the stated directions.