Three distinct points A,B,C are given in the 2-dimensional coordinate plane such that the …

Question

Three distinct points $A,B,C$ are given in the $2$-dimensional coordinate plane such that the ratio of the distance of any one of them from the point $(1,0)$ to the distance from the point $(-1,0)$ is equal to $	frac13$. Then the circumcentre of the triangle $ABC$ is at the point:

Accepted Answer

For $A=(x,y)$: $3AP=AQ$ (with $P=(1,0),Q=(-1,0)$) $\Rightarrow 9AP^2=AQ^2$:
$$9[(x-1)^2+y^2]=(x+1)^2+y^2\Rightarrow8x^2-20x+8y^2+8=0\Rightarrow x^2+y^2-\frac53x+1=0.$$
All three points lie on this circle, whose centre (the circumcentre) is $\left(\tfrac54,0\right)$.

Solutions · 1