Radical axis for two intersecting circles.

Question

For two non-concentric circles the locus of centre of circle which is orthogonal to both the circles comes out to be the radical axis of the two circles. But in case the two circles are intersecting then the radical axis will be the common secant for the two circles therefore some part of the radical axis will lie inside both the circles but from any point in this part no orthogonal circle can be drawn to any of the previous two circles. Please help me understand this contradiction.

Answer 1

It is not an "if and only if" definition. It is a characterization that follows from the definition, if certain conditions are satisfied (power of point is positive).

Specifically, if $P$ is a point on the radical axis and lies outside the (2) circle(s), then the power of $P$ with respect to the (both) circle is positive $ = R^2$, and we can construct the circle of radius $R$ that is orthogonal to the (both) circle.

When $P$ is inside the circle, then the power of the point is negative, hence we cannot construct the orthogonal circle.