Question

The coordinates of the third vertex of an equilateral triangle whose two vertices are at $(3,4),(-2,3)$ are ________.

Answer 1

Let 3rd vertex of equilateral triangle is $(x,y)$.

As all sides of equilateral triangle are same so

$(2-3{)}^2+(y-4{)}^2=(x+2{)}^2+(y+3{)}^2=5^2+1^2$

Now

$(x-3{)}^2+(y-4{)}^2=(x+2{)}^2+(y+3{)}^2$

$-6x-8y+9+16=4x-6y+4+9$

$10x+2y=12$

$5x+y=6$ __(1)

And

$(x+2{)}^2+(y+3{)}^2=25+1$

$x^2+y^2+4x-6y=13$ __(2)

By (1) & (2)

$x=(1+\sqrt{3})/2$ and $(1-\sqrt{3})/2$ and $y=6-5x$