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 5 of 6: Fourth quarter: trace the segment from U to V

X=(0,1-x,0),\ Y=(1,0,x)\Longrightarrow\ M=(\tfrac12,\tfrac{1-x}{2},\tfrac{x}{2})

Detailed analysis

While $X$ is on $DA$ and $Y$ on $BB'$ , use $X=(0,1-x,0)$ and $Y=(1,0,x)$ . Thus $M=(1/2,(1-x)/2,x/2)$ , the segment from $U$ to $V=(1/2,0,1/2)$ .