Find the third vertex of an equilateral triangle given two vertices

To make the algebra more simple suppose the  2 vertices of the triangle are at  (1,2) and  (3 ,4) and the coordinates  of the third vertex is (x,y)

Then the lengths between  the 2 vertices and the third one will be equal.

so (x - 1)^2 + (y - 2)^2  = (x - 3)^2 + (y - 4)^2

this simplifies to  x + y = 5(1)

the midpoint of the  line between the 2 given vertices  is (2,3)  and the length of the line segment between (3,4) and (2,3)  =  sqrt 2.

so the line joining (x,y) and the midpoint has length   sqrt3 * sqrt2 = sqrt6

so (x - 2)^2 + (y - 3)^2  = (sqrt6)^2  = 6(2)

Substituting x = 5 - y

(5 - y - 2)^2 + (y - 3)^2 = 6
y^2 - 6y + 6 = 0

y = 4.73
and x = 5 - 4.73 = 0.27

The tir vertex is at (4.73,0.27) Answer