Problem 3

Given tetrahedron

ABCD

, let opposite edges

AB

and

CD

have lengths

a

and

b

. Their skew-line distance is

d

and their angle is

\theta

. A plane parallel to both

AB

and

CD

divides the tetrahedron into two solids. The ratio of the plane's distances from

AB

and

CD

k

. Compute the ratio of the volumes of the two solids.

Step 2 of 5: Compute the similar tetrahedron

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V_{APWX}=\left(\frac{k}{k+1}\right)^3V.

Detailed analysis

Introduce the plane through $WX$ parallel to face $BCD$ ; it meets $AB$ at $P$ . The tetrahedron $APWX$ is similar to $ABCD$ with linear scale $k/(k+1)$ , so $V_{APWX}=k^3V/(k+1)^3$ .