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
