Inversion problem for four points

Question

If $A$, $B$, $C$, $D$ are four non-coplanar points, then show, by inversion with respect to any one of those points, that the sum of any two of the products $BC\cdot AD$, $CA\cdot BD$, $AB\cdot CD$ is greater than the third.

I have shown that if $P'$ and $Q'$ are inverses of $P$ and $Q$ with respect to an origin of inversion $O$, then $\dfrac{P'Q}{PQ} = \dfrac{OP'}{OQ}$ or $\dfrac{OQ'}{OP}$, but cannot apply this to the proof.

Any help would be appreciated.

Answer 1

Take any point,say D, and invert with radius so that A,B,C are inside the sphere. Then,say, B'C'= r^2 BC/DB.DC.The inverse points are not collinear, so, say, A'B' + B'C' is greater than A'C'. Use the foregoing expression for each term,and an easy simplification gives the required proof.