An equilateral triangle lies with its three vertices at respective distances from the origin as 1, 2 and 3 units. What is the length of each side of the triangle?

Let (x_1,y_1), (x_2,y_2), and (x_3,y_3) be the
vertices of the equilateral triangle and s its
side length. WOLOG let (x_1,y_1) = (1,0). Then

  x_2^2 + y_2^2 = 4                       (1)

  x_3^2 + y_3^2 = 9                       (2)

  s^2 = (x_2 - 1)^2 + y_2^2               (3)

  s^2 = (x_3 - 1)^2 + y_3^2               (4)

  s^2 = (x_3 - x_2)^2 + (y_3 - y_2)^2     (5)

Combining equations 1 and 3 gives

  s^2 = 5 - 2*x_2                         (6)

Combining equations 2 and 4 gives

  s^2 = 10 - 2*x_3                        (7)

Combining equations 6 and 7 gives

  x_3 = x_2 + 5/2                         (8)

Combining equations 1, 2, 5, 6, and 8 gives

  (x_2 + 1)^2*(2*x_2 - 5) = 0             (9)

The value x_2 = 5/2 does not satisfy
equation (1). Therefore,

  x_2 = -1

     and

  s = sqrt(7)

Posted by Bractals on 2007-09-05 10:43:23