alternatively, the orthogonal property of the two pencils follows from the defining property of the radical axis, that from any point "x" on the radical axis of a pencil "p" the lengths of the tangents from "x" to each circle in "p" are all equal. it follows from this that the circle centered at "x" with length equal to these tangents crosses all circles of "p" perpendicularly. the same construction can be applied for each "x" on the radical axis of "p", forming another pencil of circles perpendicular to "p".