Determining The Coordinates Of A Point Based On Its Known Difference From Three Other Points

Question

I have the coordinates of three points on a plane. Let's call them X1,Y1, X2,Y2, X3 Y3.

I need to calculate X4,Y4 but all I know is:

X1,Y1 is 350 units in distance from X4,Y4 X2,Y2 is 200 units in distance from X4,Y4 X3,Y3 is 50 units in distance from X4,Y4

I Know The Exact Values For X1,Y1, X2,Y2, and X3,Y3

How can I determine the exact location of X4,Y4?

Answer 1

(x - x1)^2 + (y - y1)^2 = r1^2  ------ p
(x - x2)^2 + (y - y2)^2 = r2^2  ------ q
(x - x3)^2 + (y - y3)^2 = r3^2  ------ r

Solve for intersection point of these 3 circles.

 p - q     ----- l 
 p - r     ----- n

Solve equation (l) and (n) using Cramer's rule.

GET_POINT(x1,y1,r1,x2,y2,r2,x3,y3,r3):
    A = x1 - x2
    B = y1 - y2
    D = x1 - x3
    E = y1 - y3

    T = (r1*r1 - x1*x1 - y1*y1)
    C = (r2*r2 - x2*x2 - y2*y2) - T
    F = (r3*r3 - x3*x3 - y3*y3) - T

    A x + B y = C/2  // this is equation 'l'
    D x + E y = F/2  // this is equation 'n'

    // Cramer's Rule

    Mx = (C E  - B F) /2
    My = (A F  - D C) /2
    M  = AE - DB

    x = Mx/M
    y = My/M

    return (x,y)

Answer 2

You post was only tagged "geometry".

A geometric solution for your problem would be to draw circles around (x1,y1), (x2,y2) and (x3,y3) with the corresponding distance to (x4,y4) as radius. (x4,y4) is the point where all thee circles intersect.