Let the coordinates of the third point be (x, y) . Since the triangle is equilateral...
distance between (x, y) and (-1, 2) = distance between (x, y) and (4, 2)
\(\sqrt{(-1-x)^2+(2-y)^2}\,=\,\sqrt{(4-x)^2+(2-y)^2} \\ (-1-x)^2+(2-y)^2\,=\,(4-x)^2+(2-y)^2 \\ (-1-x)^2\,=\,(4-x)^2 \\ (-1-x)\,=\,\pm(4-x)\\ \begin{array}\ (-1-x)=+(4-x)\qquad&\text{or}&\qquad(-1-x)=-(4-x)\\ -1-x=4-x&&\qquad-1-x=-4+x\\ -1=4&&\qquad-1=-4+2x\\ \text{not a solution}&&\qquad3=2x\\ &&\qquad\frac32=x \end{array}\)
So we know that the x coordinate must be 3/2 , which is 1.5 .
To find the y coordinate, we need to make another equation.
distance between (-1,2) and (4, 2) = 4 - -1 = 5 So...
distance between (-1, 2) and (1.5, y) = 5
\(\sqrt{(-1-1.5)^2+(2-y)^2}=5\\ \sqrt{6.25+(2-y)^2}=5\\ 6.25+(2-y)^2=25\\(2-y)^2=18.75\\2-y=\pm\sqrt{18.75}\\ -y=\pm\sqrt{18.75}-2\\ y=\pm\sqrt{18.75}+2\\ \begin{array}\ y=\sqrt{18.75}+2\qquad\text{or}\qquad&&y=-\sqrt{18.75}+2\\ y\approx6.3&&y\approx-2.3 \end{array}\)
So the solutions for (x, y) are: (1.5, 6.3) and (1.5, -2.3)
Let the coordinates of the third point be (x, y) . Since the triangle is equilateral...
distance between (x, y) and (-1, 2) = distance between (x, y) and (4, 2)
\(\sqrt{(-1-x)^2+(2-y)^2}\,=\,\sqrt{(4-x)^2+(2-y)^2} \\ (-1-x)^2+(2-y)^2\,=\,(4-x)^2+(2-y)^2 \\ (-1-x)^2\,=\,(4-x)^2 \\ (-1-x)\,=\,\pm(4-x)\\ \begin{array}\ (-1-x)=+(4-x)\qquad&\text{or}&\qquad(-1-x)=-(4-x)\\ -1-x=4-x&&\qquad-1-x=-4+x\\ -1=4&&\qquad-1=-4+2x\\ \text{not a solution}&&\qquad3=2x\\ &&\qquad\frac32=x \end{array}\)
So we know that the x coordinate must be 3/2 , which is 1.5 .
To find the y coordinate, we need to make another equation.
distance between (-1,2) and (4, 2) = 4 - -1 = 5 So...
distance between (-1, 2) and (1.5, y) = 5
\(\sqrt{(-1-1.5)^2+(2-y)^2}=5\\ \sqrt{6.25+(2-y)^2}=5\\ 6.25+(2-y)^2=25\\(2-y)^2=18.75\\2-y=\pm\sqrt{18.75}\\ -y=\pm\sqrt{18.75}-2\\ y=\pm\sqrt{18.75}+2\\ \begin{array}\ y=\sqrt{18.75}+2\qquad\text{or}\qquad&&y=-\sqrt{18.75}+2\\ y\approx6.3&&y\approx-2.3 \end{array}\)
So the solutions for (x, y) are: (1.5, 6.3) and (1.5, -2.3)