# Given a triangle ABC, on which side of the triangle does the point D lie?

• I
AD = AB + BC, BD = AC + BD, CD = AB + AC.Their various cross products will be:AD = BC, BD = AB, CD = AC.

So I've got the following problem:

I have points A, B, and C which form a triangle in a 3D space (each point of the triangle has x,y, and z coordinates). I need to find out on which side of the triangle point D lies. I do not have access to the normal of the triangle.

How am I supposed to solve this problem?

If you have the nine coordinates, then you have the plane and therewith the normal. If not, what do you have?

• topsquark
fresh_42 said:
If you have the nine coordinates, then you have the plane and therewith the normal. If not, what do you have?
I am not given any coordinates for the problem, actually, I am asking what kind of information I need so that I can solve the problem. Like a general way of solving it if that makes sense. After I have the normal how would I approach the question?

By the right-hand rule or something. In order to determine a side you must first have criteria to distinguish them. Two vectors spanning the plane, plus the normal give you either the right or left hand.

• The coordinates of the four points, ##A, B, C, D##, is all the information you need. You first must define which "side" is which.
Suppose you define the side by the cross product between the ##V_1 = \bar{AB}## and ##V_2 = \bar{AC}## vectors. Using the right-hand rule, form the cross product ## N=V_1 \times V_2##. One side would be those points, ##X##, where the vector from ##A## to ##X## has a positive dot product with ##N##. The other side would have a negative dot product with ##N##.

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• fresh_42 said:
If you have the nine coordinates, then you have the plane
And three lines. Plug and check.
I am not given any coordinates for the problem
What then does it mean to say you have three points? What do you have about them if not their location?

• topsquark
There are two ways of interpreting "side of the triangle".
One is which side of the triangles plane does point D land.
The other is which of the three sides of the triangle does point D land.

Go with @FactChecker for the first interpretation.

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