If `(x_i ,y_i),i=1,2,3,` are the vertices of an equilateral triangle such that `(x_1+2)^2+(y_1-3)^2=(x_2+2)^2+(y_2-3)^2=(x_3+2)^2+(y_2-3)^2=(x_3+2)^2+(y_3-3)^2,` then find the value of `(x_1+x_2+x_3)/(y_1+y_2+y_3)` .

Answer

`-2//3`

Explanation

`(x_1+2)^2+(y_1-3)^2=(x_2+2)^2+(y_2-3)^2=(x_3+2)^2+(y_3-3)^2`
Therefore,the circumcenter of the triangle formed by points `(x_1,y_1),(x_2,y_2,(x_3,y_3)` is `(-2,3)`. But the triangle is equilateral, so,that centroid is `(-2,3)`.Therefore,
`(x_1+x_2+x_3)/(3)=-2,(y_1+y_2+y_3)/(3)=3`
or `(x_1+x_2+x_3)/(y_1+y_2+y_3)=-(2)/(3)`