MathLabs

Problem 3

Given tetrahedron ABCDABCD, let opposite edges ABAB and CDCD have lengths aa and bb. Their skew-line distance is dd and their angle is θ\theta. A plane parallel to both ABAB and CDCD divides the tetrahedron into two solids. The ratio of the plane's distances from ABAB and CDCD is kk. Compute the ratio of the volumes of the two solids.
Step 1 of 5: Find the common section ratio
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In plain words

Parallel slices preserve one affine scale factor along every edge.

AXXD=BYYD=BZZC=AWWC=k1,qquadAXAD=kk+1.\frac{AX}{XD}=\frac{BY}{YD}=\frac{BZ}{ZC}=\frac{AW}{WC}=\frac{k}{1},\\qquad \frac{AX}{AD}=\frac{k}{k+1}.
A tetrahedron in the parallel-plane setup
A translucent 3D tetrahedron illustrating the spatial setting for a plane parallel to two opposite edges.
Detailed analysis

Let the plane meet AD,BD,BC,ACAD,BD,BC,AC at X,Y,Z,WX,Y,Z,W. Because the plane is parallel to both opposite edges, distances along transversals vary linearly; the given distance ratio gives AX:XD=k:1AX:XD=k:1. The same fraction holds on every edge from the corresponding vertex.