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 1 of 5: Choose coordinates for the cube

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In plain words

The two free positions are independent parameters, so the locus is the image of a parameter square.

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

A cube for orienting the two parallel faces and their diagonals.

Detailed analysis

Take the cube edge as $1$ , put $ABCD$ in the plane $z=1$ , and put $A'B'C'D'$ in $z=0$ . Then $X$ and $Y$ lie on diagonals of the two parallel faces, which makes both locus questions affine-coordinate calculations.