Power of a point

Besides the properties mentioned in the lead there are further properties: Orthogonal circle For any point P outside of the circle c there are two tangent points T_1,T_2 on circle c, which have equal distance to P. Hence the circle o with center P through T_1 passes T_2, too, and intersects c orthogonal: • The circle with center P and radius \sqrt{\Pi(P)} intersects circle c orthogonal. If the radius \rho of the circle centered at P is different from \sqrt{\Pi(P)} one gets the angle of intersection \varphi between the two circles applying the Law of cosines (see the diagram): :\rho^2+r^2-2\rho r \cos\varphi=|PO|^2 : \rightarrow \ \cos\varphi=\frac{\rho^2+r^2-|PO|^2}{2\rho r}=\frac{\rho^2-\Pi(P)}{2\rho r} (PS_1 and OS_1 are normals to the circle tangents.) If P lies inside the blue circle, then \Pi(P) and \varphi is always different from 90^\circ. If the angle \varphi is given, then one gets the radius \rho by solving the quadratic equation :\rho^2-2\rho r \cos\varphi-\Pi(P)=0. Intersecting secants theorem, intersecting chords theorem For the intersecting secants theorem and chord theorem the power of a point plays the role of an invariant: • Intersecting secants theorem: For a point P outside a circle c and the intersection points S_1,S_2 of a secant line g with c the following statement is true: |PS_1| \cdot |PS_2|= \Pi(P), hence the product is independent of line g. If g is tangent then S_1=S_2 and the statement is the tangent-secant theorem. • Intersecting chords theorem: For a point P inside a circle c and the intersection points S_1,S_2 of a secant line g with c the following statement is true: |PS_1| \cdot |PS_2|= -\Pi(P), hence the product is independent of line g. Radical axis Let P be a point and c_1,c_2 two non concentric circles with centers O_1,O_2 and radii r_1,r_2. Point P has the power \Pi_i(P) with respect to circle c_i. The set of all points P with \Pi_1(P)=\Pi_2(P) is a line called radical axis. It contains possible common points of the circles and is perpendicular to line \overline{O_1O_2}. == Secants theorem, chords theorem: common proof ==

Both theorems, including the tangent-secant theorem, can be proven uniformly: Let P:\vec p be a point, c: \vec x^2-r^2=0 a circle with the origin as its center and \vec v an arbitrary unit vector. The parameters t_1,t_2 of possible common points of line g: \vec x=\vec p+t\vec v (through P) and circle c can be determined by inserting the parametric equation into the circle's equation: :(\vec p+t\vec v)^2-r^2=0 \quad \rightarrow \quad t^2+2t\;\vec p\cdot\vec v +\vec p^2-r^2=0 \ . From Vieta's theorem one finds: :t_1\cdot t_2=\vec p^2-r^2 =\Pi(P). (independent of \vec v) \Pi(P) is the power of P with respect to circle c. Because of |\vec v|=1 one gets the following statement for the points S_1,S_2: :|PS_1|\cdot|PS_2|=t_1t_2=\Pi(P)\ , if P is outside the circle, :|PS_1|\cdot|PS_2|=-t_1t_2=-\Pi(P)\ , if P is inside the circle (t_1,t_2 have different signs !). In case of t_1=t_2 line g is a tangent and \Pi(P) the square of the tangential distance of point P to circle c. == Similarity points, common power of two circles ==

Similarity points Similarity points are an essential tool for Steiner's investigations on circles. Given two circles :\ c_1: (\vec x -\vec m_1)^2-r_1^2=0, \quad c_2: (\vec x -\vec m_2)^2-r_2^2=0 \ . A homothety (similarity) \sigma, that maps c_1 onto c_2 stretches (jolts) radius r_1 to r_2 and has its center Z:\vec z on the line \overline{M_1 M_2}, because \sigma(M_1)=M_2. If center Z is between M_1,M_2 the scale factor is s=-\tfrac{r_2}{r_1}. In the other case s=\tfrac{r_2}{r_1}. In any case: :\sigma(\vec m_1)=\vec z + s(\vec m_1-\vec z)=\vec m_2. Inserting s=\pm\tfrac{r_2}{r_1} and solving for \vec z yields: : \vec z= \frac{r_1\vec m_2\mp r_2\vec m_1}{r_1\mp r_2}. Point E:\vec e=\frac{r_1\vec m_2-r_2\vec m_1}{r_1-r_2} is called the exterior similarity point and I:\vec i=\frac{r_1\vec m_2+r_2\vec m_1}{r_1+r_2} is called the inner similarity point. In case of M_1=M_2 one gets E=I=M_i. In case of r_1=r_2: E is the point at infinity of line \overline{M_1 M_2} and I is the center of M_1,M_2. In case of r_1=|EM_1| the circles touch each other at point E inside (both circles on the same side of the common tangent line). In case of r_1=|IM_1| the circles touch each other at point I outside (both circles on different sides of the common tangent line). Further more: • If the circles lie disjoint (the discs have no points in common), the outside common tangents meet at E and the inner ones at I. • If one circle is contained within the other, the points E,I lie within both circles. • The pairs M_1,M_2;E,I are projective harmonic conjugate: Their cross ratio is (M_1,M_2;E,I)=-1. Monge's theorem states: The outer similarity points of three disjoint circles lie on a line. Common power of two circles Let c_1,c_2 be two circles, E their outer similarity point and g a line through E, which meets the two circles at four points G_1,H_1,G_2,H_2. From the defining property of point E one gets :\frac=\frac{r_1}{r_2}=\frac\ : \rightarrow \ |EG_1|\cdot|EH_2|=|EH_1|\cdot|EG_2|\ and from the secant theorem (see above) the two equations :|EG_1|\cdot|EH_1|=\Pi_1(E),\quad |EG_2|\cdot|EH_2|=\Pi_2(E) . Combining these three equations yields: \begin{align} \Pi_1(E)\cdot\Pi_2(E) &=|EG_1|\cdot|EH_1|\cdot|EG_2|\cdot|EH_2| \\ &=|EG_1|^2\cdot|EH_2|^2= |EG_2|^2\cdot|EH_1|^2 \ . \end{align} Hence: |EG_1|\cdot|EH_2|= |EG_2| \cdot |EH_1|=\sqrt{ \Pi_1(E)\cdot\Pi_2(E)} (independent of line g !). The analog statement for the inner similarity point I is true, too. The invariants \sqrt{\Pi_1(E)\cdot\Pi_2(E)},\ \sqrt{ \Pi_1(I)\cdot\Pi_2(I)} are called by Steiner common power of the two circles (gemeinschaftliche Potenz der beiden Kreise bezüglich ihrer Ähnlichkeitspunkte). The pairs G_1,H_2 and H_1,G_2 of points are antihomologous points. The pairs G_1,G_2 and H_1,H_2 are homologous. Determination of a circle that is tangent to two circles For a second secant through E: :|EH_1|\cdot|EG_2|= |EH'_1|\cdot|EG'_2| From the secant theorem one gets: :The four points H_1,G_2,H'_1,G'_2 lie on a circle. And analogously: : The four points G_1,H_2,G'_1,H'_2 lie on a circle, too. Because the radical lines of three circles meet at the radical (see: article radical line), one gets: :The secants \overline{H_1H'_1},\;\overline{G_2G'_2} 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 \overline{M_1H_1},\overline{M_2G_2}. The secants \overline{H_1H'_1}, \overline{G_2G'_2} become tangents at the points H_1,G_2. The tangents intercept at the radical line p (in the diagram yellow). Similar considerations generate the second tangent circle, that meets the given circles at the points G_1,H_2 (see diagram). All tangent circles to the given circles can be found by varying line g. ;Positions of the centers If X is the center and \rho the radius of the circle, that is tangent to the given circles at the points H_1,G_2, then: :\rho=|XM_1|-r_1=|XM_2|-r_2 : \rightarrow \ |XM_2|-|XM_1|=r_2-r_1 . Hence: the centers lie on a hyperbola with :foci M_1,M_2, :distance of the vertices 2a=r_2-r_1, :center M is the center of M_1,M_2 , :linear eccentricity c=\tfrac{2} and :\ b^2=e^2-a^2=\tfrac{|M_1M_2|^2-(r_2-r_1)^2}{4}. Considerations on the outside tangent circles lead to the analog result: If X is the center and \rho the radius of the circle, that is tangent to the given circles at the points G_1,H_2, then: :\rho=|XM_1|+r_1=|XM_2|+r_2 : \rightarrow \ |XM_2|-|XM_1|=-(r_2-r_1) . The centers lie on the same hyperbola, but on the right branch. See also Problem of Apollonius. == Power with respect to a sphere ==

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. ==Darboux product==

The power of a point is a special case of the Darboux product between two circles, which is given by :\left| A_1A_2 \right|^2-r_1^2-r_2^2 \, where A1 and A2 are the centers of the two circles and r1 and r2 are their radii. The power of a point arises in the special case that one of the radii is zero. If the two circles are orthogonal, the Darboux product vanishes. If the two circles intersect, then their Darboux product is :2r_1 r_2 \cos\varphi \, where φ is the angle of intersection (see section orthogonal circle). ==Laguerre's theorem==

Laguerre defined the power of a point P with respect to an algebraic curve of degree n to be the sum 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 . 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. ==References==