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.