# Find P points inside ABCD tetrahedron so that volume of ABCP = volume of ABDP

Find all $P$ points inside $ABCD$ tetrahedron, so that $V_{ABCP} = V_{ABDP}$

Thanks in advance for any help.

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Express the volumes: $$V[ABCP]=\frac{1}{3} d(P,ABC) Area(ABC),\ V[ABDP]=\frac{1}{3} d(P,ABD) Area(ABD)$$
The equality of the two volumes prove that $$\frac{d(P,ABC)}{d(P,ABD)}=\frac{Area(ABC)}{Area(ABD)}$$
So the distances from $P$ to the planes $ABC$ and $ABD$ are proportional. I think the locus is a plane, but try and see it for yourself.
Well, first the plane must contain $AB$, and one point on it must satisfy the ratio constraint. You can pick that point to be on $CD$, for example. So your locus will be a triangle $ABX$ with $X \in CD$ with the corresponding ratio. – Beni Bogosel Apr 6 '12 at 11:03