Take the 2-minute tour ×
Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. It's 100% free, no registration required.

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

Thanks in advance for any help.

share|improve this question

1 Answer 1

up vote 2 down vote accepted

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.

share|improve this answer
    
Yes it must be a plane, but how can I specify which plane, if say the areas of the sides arent equal? –  Gabor Magyar Apr 6 '12 at 10:08
    
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

Your Answer

 
discard

By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.