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 2 of 6: First quarter: trace the segment from V to W

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X=(x,0,0),\ Y=(1,x,1)\Longrightarrow\ M=(\tfrac{x+1}{2},\tfrac{x}{2},\tfrac12),\quad0\le x\le1

Detailed analysis

While $X$ moves on $AB$ and $Y$ on $B'C'$ , synchronized equal speeds give $X=(x,0,0)$ and $Y=(1,x,1)$ . Their midpoint is $M=(x/2+1/2,x/2,1/2)$ , the segment joining $V=(1/2,0,1/2)$ to $W=(1,1/2,1/2)$ .