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 6 of 6: Identify the closed locus as a rhombus

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U=(\tfrac12,\tfrac12,0),\quad V=(\tfrac12,0,\tfrac12),\quad W=(1,\tfrac12,\tfrac12)\quad\Longrightarrow\quad\text{locus}=CUVW

Detailed analysis

The four traced segments are $VW$ , $WC$ , $CU$ , and $UV$ . Their four vertices are the centers of $ABCD$ , $ABB'A$ , and $BCC'B'$ , together with $C$ ; all four sides have equal length, so the locus is the rhombus $CUVW$ .