Problem 5

The cube

ABCDA'B'C'D'

has

A

above

A'

B

above

B'

, and so on. Let

X

be any point of the face diagonal

AC

and

Y

any point of

B'D'

. (a) Find the locus of the midpoint of

XY

. (b) Find the locus of the point

Z

XY

such that

ZY=2XZ

Step 2 of 5: Parameterize the two diagonal points

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X=(t,t,1),\quad Y=(1-s,s,0),\quad 0\le t,s\le1

Detailed analysis

Every point of $AC$ has the form $X=(t,t,1)$ and every point of $B'D'$ has the form $Y=(1-s,s,0)$ , with $0\le t,s\le1$ . The parameters vary independently over a square.