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# Triangulation equation?

I have three points in an X,Y coordinate system. Those points are A, B, and C forming a triangle. I want to know the coordinates of A. The coordinates of B and C, the distance AB, and the angle A between AB and AC are all known.

### 2 Answers

- DaveLv 67 years agoFavorite Answer
By Law of Cosines

(BC)^2=(AB)^2+(AC)^2 -2(AC)(AB)cos(<A)

We know AB, the angle A, and we can easily find BC knowing the coordinates.

solving the quadratic equation,

0=(AC)^2 -2(AB)cos(<A)(AC)+((AB)^2-(BC)^2)

we know the distance AC.

(AB)^2=(Bx-Ax)^2+(By-Ay)^2

(AC)^2=(Cx-Ax)^2+(Cy-Ay)^2

using these two equations, we can easily solve for Ax and Ay.

- Let'squestionLv 77 years ago
Let B (X₂, Y₂), C (X₃, Y₃); let the given angle A be θ degree and distance AB = d. Point A(h, k) or we need find h and k. For this- We need to solve following two simultaneous equations .

([{Y₂-k)/(X₂-h)} - {Y₃-k)/(X₃-h)}]/[1+({Y₂-k)/(X₂-h)}*{Y₃-k)/(X₃-h)}]) = tan θ ------------ (1) and

(k-Y₂)² + (h-X₂)² = d² --------------------------------------------------------------------------------------(2)