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 5 of 5: Identify the rectangle and its dimensions

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Z_x+Z_y=\dfrac{4t+1}{3},\quad Z_x-Z_y=\dfrac{1-2s}{3}

Detailed analysis

Now $Z_x+Z_y$ ranges over $[1/3,5/3]$ and $Z_x-Z_y$ over $[-1/3,1/3]$ . The two parameter directions are perpendicular, with lengths $2\sqrt2/3$ and $\sqrt2/3$ . Thus the locus is a filled rectangle in $z=2/3$ , of dimensions $2\sqrt2/3\times\sqrt2/3$ .