Third point of a triangle from only two points and all three edge lengths

Question

I want a triangle composed of points A, B and C in Cartesian 3D space.

I currently know the positions of points A and B, but I need point C. I have the line segment AB, and thus its magnitude. I have only the magnitudes of line segments AC and BC.

From this data, how do I derive point C? Please explain your logic. Thank you for your help.

If I understand the question correctly, it is not uniquely determined. — Pandora, Mar 25 '11 at 17:32
There are an infinite amount of possible point C's. Imagine the intersection of two spheres of radius $|AC|$ coming from point $A$ and radius $|BC|$ coming from point $B$. — Justin L., Mar 25 '11 at 17:33
its only defined up to a rotation...you can use the law of cosines to get the angles — yoyo, Mar 25 '11 at 17:34

score 2 · Answer 1 · answered Mar 25 '11 at 17:36

2

The lengths of segments AB, AC and BC are not enough to uniquely identify a triangle in 3-d space.

You can just pick a random plane that contains AB and make your triangle there.

answered Mar 25 '11 at 17:36

Eelvex

1,414

2

Even on a plane, there are two such points for C, and so two triangles. – Henry Mar 25 '11 at 17:43
@Henry: yes, but then it's just a matter of sign-choosing. – Eelvex Mar 25 '11 at 17:48

score 2 · Answer 2 · answered Mar 26 '11 at 12:18

2

The locus of point C is a circle (with a center inside AB) or nothing (depends on magnitudes of segments AC and BC).

answered Mar 26 '11 at 12:18

saeedgnu

183

Third point of a triangle from only two points and all three edge lengths

2 Answers2