Demonstration of Common Elements of Involution on a Simple Example

The involution of projective rows with a common support, its geometric interpretation are considered. Taking the special case of the geometric interpretation of involution, the problem of constructing harmonically conjugate points is solved for given initial conditions, when one circle and a radical axis of this circle with a bundle of corresponding circles with a common radical axis are given. A proposal is given on the existence of a single circle in a bundle, the diametrical points of which on the lines of centers make up a harmonic four with diametral points of a given circle. It is shown that using the diametrical points of a given circle and points P, Q of the radical axis in elliptical involution, you can build double points X, Y and the radical axis of the PQ of circles in hyperbolic involution. And the tangent from the vertical diammetral point of the circle w1 to the circle passing through double points of hyperbolic involution - there is a point P(Q) of the radical axis of elliptical involution. The indicated properties make it possible to carry out a mutual transition from one involution to another. It was established that the diagonals of the quadrangles obtained when crossing all the circles of the bundle, orthogonal to the two given in elliptical involution, intersect in the center of the radical axis of the given circles in hyperbolic involution, and the diagonals of the quadrangles of all circles of the beam in hyperbolic involution are intersected in the center of the radical axis of the given circles in elliptical Involution. The geometric place (GP) of each point of the harmonic four is constructed. In this case, the geometric place a pair of harmonic four in an elliptic involution turns out to be an ellipse that has a common tangent at points P with the circle of double points of the hyperbolic involution. And the GP pairs of the harmonic four for hyperbolic involution are two branches of the hyperbola that pass through the centers of the circles that define the elliptical involution.