Fermat-Apollonius circle

The Fermat-Apollonius circle of a triangle is a special circle related to ratios of distances from a point to the triangle's vertices. Given triangle $ABC$ and a positive real number $k \neq 1$, the Apollonius circle with respect to $A$ is the set of all points $P$ in the plane such that $\frac{PB}{PC} = k$. The Fermat-Apollonius circle is the unique Apollonius circle that passes through the triangle's Fermat point (the point minimizing the total distance to $A$, $B$, and $C$).

Important properties

• The Fermat-Apollonius circle is an Apollonius circle associated with a triangle.

• It passes through the Fermat point of the triangle.

• It is defined by a specific ratio of distances from a point to two vertices of the triangle.

• The center and radius of the circle depend on the triangle and the chosen ratio.