intersecting a conic at points {A',B'} and {C',D'} correspondingly. Then the ratio of the products k =

(PA'*PB')/(PC'*PD') is independent of the position of point P.

The proof (taken from [Loney, p. 371]) is a calculation using the general equation of the conic.

Obviously last expression is independent from the position (x',y') of P.

squares of the diameters which are parallel to AB and CD.

circle, therefore the name on the title of this section. In fact, if the conic is a circle then k=1 for all

possible directions. The inverse is also true: If k=1 for all possible directions then the conic is a

circle. It even suffices to check if k=1 for three pairs of three different directions.

{AB,CD} both intersect the conic at two real points. The cases for which one of the secants intersects the

conic at one only point (can occure in parabolas and/or hyperbolas) is discussed in the files

PowerGeneralParabola.html and PowerGeneralHyperbola.html respectively.

diameters which are parallel to these tangents.

Follows from the previous discussion and remark by taking the special position of P, so that the

secants become tangents.

P'. Then the ratio

k = (PA

of P and P'.

The proof follows from the initial calculation by dividing the corresponding ratios for the two points

P(x',y'), P'(x'',y''), the polar angle being the same:

inclined to the axes of the conic.

By the initial proposition (EA*EB)/(EC*ED) is independent from the position of E. Since the four

points are concyclic, this ratio is 1. Thus, by selecting another E so that the parallels from E to CD

and AB become tangents, we see that these two tangents must have equal lengths. This is only

possible if that E is on the axis of the conic.

two points. Then the other intersection points {C,D} of the circles define lines which are parallel to

a fixed direction.

This is immediate consequence of the previous proposition. Lines DC and AB are equal inclined to

the axes of the conic.

circles tangent to a conic at a fixed point A of the conic with tangent there L

intersection points {C,D} of the bundle members and the conic define lines CD that are parallel to

direction L' which is symmetric to L

the base points {A,B} not lying on the conic. There are though some cases in which the circles of a

bundle of non-intersecting type intersect a hyperbola along points in parallel directions (see

CircleBundleTransformationHyperbola2.html ).

In the file RectHypeParaChords.html is contained a synthetic proof of this fact for the case of

rectangular hyperbolas.

PowerGeneralParabola.html

PowerGeneralHyperbola.html

CircleBundleTransformationHyperbola2.html

RectHypeParaChords.html

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