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 3 of 5: Find the midpoint locus

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M=\dfrac{X+Y}{2}=\left(\dfrac{t+1-s}{2},\dfrac{t+s}{2},\dfrac12\right)

Detailed analysis

The midpoint $M$ always has $z=1/2$ . Its other coordinates are affine in $t,s$ . More revealingly, $M_x+M_y=1/2+t$ ranges over $[1/2,3/2]$ , while $M_x-M_y=1/2-s$ ranges over $[-1/2,1/2]$ . Thus $M$ fills a square in the mid-plane, with side length $1/\sqrt2$ , whose vertices are the midpoints of the four vertical faces.