Intersecting secants theorem: For a point outside a circle and the intersection points of a secant line with the following statement is true: , hence the product is independent of line . If is tangent then and the statement is the tangent-secant theorem.
Intersecting chords theorem: For a point inside a circle and the intersection points of a secant line with the following statement is true: , hence the product is independent of line .
Let be a point and two non concentric circles with
centers and radii . Point has the power with respect to circle . The set of all points with is a line called radical axis. It contains possible common points of the circles and is perpendicular to line .
Secants theorem, chords theorem: common proof
Both theorems, including the tangent-secant theorem, can be proven uniformly:
Let be a point, a circle with the origin as its center and an arbitrary unit vector. The parameters of possible common points of line (through ) and circle can be determined by inserting the parmetric equation into the circle's equation:
Similarity points are an essential tool for Steiner's investigations on circles.
Given two circles
A homothety (similarity) , that maps onto stretches (jolts) radius to and has its center on the line , because . If center is between the scale factor is . In the other case . In any case:
Inserting and solving for yields:
Similarity points of two circles: various cases
is called the exterior similarity point and
is called the inner similarity point.
In case of one gets .
In case of : is the point at infinity of line and is the center of .
In case of the circles touch each other at point inside (both circles on the same side of the common tangent line).
In case of the circles touch each other at point outside (both circles on different sides of the common tangent line).
If the circles lie disjoint (the discs have no points in common), the outside common tangents meet at and the inner ones at .
If one circle is contained within the other, the points lie within both circles.
Similarity points of two circles and their common power
Let be two circles, their outer similarity point and a line through , which meets the two circles at four points . From the defining property of point one gets
and from the secant theorem (see above) the two equations
Combining these three equations yields:
(independent of line !).
The analog statement for the inner similarity point is true, too.
The invariants are called by Steiner common power of the two circles (gemeinschaftliche Potenz der beiden Kreise bezüglich ihrer Ähnlichkeitspunkte).
The pairs and of points are antihomologous points. The pairs and are homologous.
Determination of a circle that is tangent to two circles
Common power of two circles: application
Circles tangent to two circles
For a second secant through :
From the secant theorem one gets:
The four points lie on a circle.
The four points lie on a circle, too.
Because the radical lines of three circles meet at the radical (see: article radical line), one gets:
The secants meet on the radical axis of the given two circles.
Moving the lower secant (see diagram) towards the upper one, the red circle becomes a circle, that is tangent to both given circles. The center of the tangent circle is the intercept of the lines . The secants become tangents at the points . The tangents intercept at the radical line (in the diagram yellow).
Similar considerations generate the second tangent circle, that meets the given circles at the points (see diagram).
All tangent circles to the given circles can be found by varying line .
Positions of the centers
Circles tangent to two circles
If is the center and the radius of the circle, that is tangent to the given circles at the points , then:
The idea of the power of a point with respect to a circle can be extended to a sphere
. The secants and chords theorems are true for a sphere, too, and can be proven literally as in the circle case.
Laguerre defined the power of a point P with respect to an algebraic curve of degree n to be the product of the distances from the point to the intersections of a circle through the point with the curve, divided by the nth power of the diameter d. Laguerre showed that this number is independent of the diameter (Laguerre 1905). In the case when the algebraic curve is a circle this is not quite the same as the power of a point with respect to a circle defined in the rest of this article, but differs from it by a factor of d2.
^Michel Chasles, C. H. Schnuse: Die Grundlehren der neuern Geometrie, erster Theil, Verlag Leibrock, Braunschweig, 1856, p. 312
^ William J. M'Clelland: A Treatise on the Geometry of the Circle and Some Extensions to Conic Sections by the Method of Reciprocation,1891, Verlag: Creative Media Partners, LLC,ISBN: 9780344903748, p. 121,220
^K.P. Grothemeyer: Analytische Geometrie, Sammlung Göschen 65/65A, Berlin 1962, S. 54
^Pierre Larochelle, J. Michael McCarthy:Proceedings of the 2020 USCToMM Symposium on Mechanical Systems and Robotics, 2020, Springer-Verlag, ISBN 978-3-030-43929-3, p. 97
Johnson RA (1960), Advanced Euclidean Geometry: An elementary treatise on the geometry of the triangle and the circle (reprint of 1929 edition by Houghton Miflin ed.), New York: Dover Publications, pp. 28–34, ISBN978-0-486-46237-0