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Power of a point

In elementary plane geometry, the power of a point P with respect to a circle with center O and radius r is defined as the real number PO^2 - r^2, which quantifies the relative position of P to the circle: positive if P is outside, zero if on the circle, and negative if inside.^[1] This value equals the square of the length of the tangent from P to the circle when P is outside, and for any line through P intersecting the circle at points A and B, it equals the product PA \cdot PB.^[2] The power of a point theorem states that if two lines through P intersect the circle at A, B and C, D respectively, then PA \cdot PB = PC \cdot PD, a constant equal to the power; this holds for intersecting chords (when P is inside), secants (when outside), or one tangent and one secant.^[1] The converse asserts that if PA \cdot PB = PC \cdot PD for lines AB and CD intersecting at P, then A, B, C, D are concyclic.^[2] These relations, proven via similar triangles or inscribed angle theorems, extend to applications in Olympiad problems and computational geometry.^[1] Beyond single circles, the concept generalizes to the radical axis of two circles—the locus of points with equal power to both, a line perpendicular to the line joining their centers—and the radical center, the concurrency point of radical axes for three circles.^[2] The term and formal definition were introduced by Swiss mathematician Jakob Steiner in his 1826 manuscript "Einige geometrische Betrachtungen," where he linked it to loci satisfying D^2 - d^2 = constant via the Pythagorean theorem, building on earlier Euclidean propositions without using the phrase explicitly.^[3]

Definition and Interpretation

Formal Definition

In geometry, the power of a point P with respect to a circle with center O and radius r is defined as the quantity |PO|^2 - r^2.^[4] This value, often denoted p_c(P), remains constant for any line through P intersecting the circle and captures the relative position of P to the circle. Equivalently, if P lies outside the circle and a tangent from P touches the circle at point T, then the power equals the square of the tangent length, |PT|^2.^[4]^[5] The sign of the power follows a specific convention: it is positive when P is outside the circle, zero when P lies on the circle, and negative when P is inside the circle.^[4]^[5] This algebraic expression aligns with alternative geometric interpretations, such as the product of the directed lengths of segments formed by a secant line from P intersecting the circle at two points A and B, yielding PA \cdot PB (positive outside, negative inside).^[4] The term "power of a point" was coined by Jakob Steiner in 1826, in his work Einige geometrische Betrachtungen.^[6]

Geometric Interpretations

The power of a point with respect to a circle exhibits a fundamental invariance: for any fixed point P and circle, the product of the directed lengths from P to the two intersection points of any line through P with the circle remains constant, regardless of the line's direction. This property underscores the circle's rotational symmetry around its center and provides a measure of P's position relative to the circle—positive if P is outside, zero if on the circumference, and negative if inside—without depending on specific secant orientations.^[7]^[8] Geometrically, this invariance manifests in visualizations that highlight equal segment products. For a point P outside the circle, two tangent lines from P touch the circle at points T_1 and T_2, with the squared tangent length PT_1 = PT_2 equaling the power; any secant through P intersecting at A and B then satisfies PA \cdot PB matching this value, illustrating consistency across configurations. Inside the circle, chords through P intersecting at A and B yield PA \cdot PB equal to the power (negative), using directed segments to account for the interior position, where no real tangents exist but the product remains uniform for all such chords. These diagrams emphasize the power as a scalar invariant tying diverse line-circle intersections to a single geometric essence.^[7] In inversive geometry, the power connects directly to inversion transformations, where a point P inverts to P' such that the product of distances from the inversion center equals the square of the inversion radius; inversion maps circles through the center to straight lines, preserving angles and circles while revealing the power as a squared distance measure intrinsic to such mappings. This link extends classical Euclidean properties into a broader framework, treating lines as circles through infinity and using the power to quantify inversive distances between circles.^[7]^[9]

Core Theorems

Intersecting Chords Theorem

The intersecting chords theorem applies when a point lies inside a circle. Consider a circle with two chords, AB and CD, that intersect at a point P within the circle, where A, B, C, and D are distinct points on the circumference. The theorem states that the product of the lengths of the segments of one chord equals the product of the lengths of the segments of the other chord: PA \cdot PB = PC \cdot PD.^[10] This configuration forms a diagram of crossing chords dividing the circle into four arcs, with P as the intersection point and the segments labeled as PA, PB on chord AB and PC, PD on chord CD. This equality represents a specific case of the power of a point with respect to the circle, where the common product PA \cdot PB = PC \cdot PD equals the power of P, a fixed value depending only on P's position relative to the circle.^[11] The theorem assumes the intersection occurs inside the circle, distinguishing it from cases where lines intersect the circle from an external point. For example, suppose two chords intersect inside a circle such that one chord is divided into segments of lengths 5 and 12, while the adjacent segment on the other chord measures 6; the theorem implies the remaining segment on the second chord is 10, since $5 \times 12 = 6 \times 10 = 60, verifying the equal products.^[12]

Secant-Secant Theorem

The secant-secant theorem states that if two secant lines are drawn from an external point P to a circle, with one secant intersecting the circle at points A and B (where A is closer to P) and the other at points C and D (where C is closer to P), then the product of the entire length of each secant and its external part is equal:
PA \times PB = PC \times PD.
This equality holds for any pair of secants from P, reflecting the constant power of the point P with respect to the circle.^[13]^[14] In the typical diagram, point P lies outside the circle, and the two secant lines extend from P through the circle, each crossing the circumference at two distinct points: the first secant at A and B, the second at C and D, forming four points of intersection on the circle. The segments PA, PB, PC, and PD are measured along these lines, with the theorem stating that PA \times PB = PC \times PD.^[4] This theorem relates to the tangent-secant case of the power of a point, where one secant degenerates into a tangent line from P touching the circle at point T; in that limit, the power equals the square of the tangent length: PA \times PB = PT^2.^[13] For illustration, suppose two secants emanate from external point P to a circle, with the first secant having an external segment PA = 4 units and internal segment AB = 2 units (so PB = 6 units), and the second having external segment PC = 3 units and internal segment CD = 5 units (so PD = 8 units). The products are $4 \times 6 = 24 and $3 \times 8 = 24, confirming the equality.^[15] The secant-secant theorem finds applications in surveying and optics, particularly for external observations involving circular paths or lenses, where it aids in computing inaccessible distances through intersecting lines of sight.^[16]

Proofs of Core Theorems

Proof via Similar Triangles

The proof of the power of a point theorem via similar triangles relies on establishing geometric similarities in the configurations of intersecting chords and secants relative to a circle. Consider a circle with center O and a point P inside the circle. Two chords through P intersect the circle at points A and B on one chord, and C and D on the other, such that the chords are AB and CD crossing at P. This setup forms triangles \triangle APD and \triangle CPB. The angle \angle APD equals \angle CPB as they are vertical angles at the intersection point P. Additionally, \angle PAD = \angle PCB because both are inscribed angles subtending the same arc BD. Thus, by the AA similarity criterion, \triangle APD \sim \triangle CPB.^[17] From this similarity, the corresponding sides are proportional: \frac{PA}{PC} = \frac{PD}{PB}. Rearranging the proportion yields PA \cdot PB = PC \cdot PD, establishing the intersecting chords theorem. This derivation highlights the power of P with respect to the circle as the constant product of segment lengths.^[18]^[19] For the case where P lies outside the circle, extend the proof to secants. Two secants from P intersect the circle at A and B (with A closer to P) on one line, and C and D (with C closer to P) on the other. Draw chords AD and CB. The relevant triangles are \triangle PAD and \triangle PCB. The angle at P is common to both. Additionally, \angle PAD = \angle PCB because both subtend arc BD, and \angle PDA = \angle PBC because both subtend arc AC. Thus, by the AA similarity criterion, \triangle PAD \sim \triangle PCB. The similarity implies \frac{PA}{PC} = \frac{PD}{PB}. Cross-multiplying gives PA \cdot PB = PC \cdot PD, proving the secant-secant theorem. This step-by-step angle chasing—identifying equal angles via arc subtensions and applying proportion rules—directly links the external point's segments to the power invariant.^[18] This synthetic approach using similar triangles is elementary, requiring only basic properties of circles and Euclidean geometry without coordinates, trigonometry, or algebraic manipulations, making it accessible for introductory proofs.^[19]

Proof via Inscribed Angles

The proof of the intersecting chords theorem via inscribed angles relies on the fundamental property that inscribed angles subtending the same arc are equal, combined with the law of sines in appropriate triangles. Consider two chords AB and CD intersecting at point P inside the circle. The vertical angles at P are equal by the vertical angles theorem, but the key equalities arise from inscribed angles: specifically, ∠APD and ∠BPC are vertical, while ∠PAD equals ∠PCB as both subtend arc BD, and ∠ADP equals ∠CBP as both subtend arc AC. To derive the segment product equality without directly invoking similarity, apply the law of sines to triangles APD and CPB. In △APD, \frac{AP}{\sin \angle ADP} = \frac{PD}{\sin \angle PAD}. Similarly, in △CPB, \frac{CP}{\sin \angle CBP} = \frac{PB}{\sin \angle PCB}. Since ∠ADP = ∠CBP and ∠PAD = ∠PCB, the sines of equal angles are equal, so the ratios simplify to \frac{AP}{PD} = \frac{CP}{PB}. Cross-multiplying yields AP \cdot PB = PD \cdot CP, establishing the theorem.^[20] For the secant-secant theorem, where two secants from external point P intersect the circle at A, B and C, D respectively (with A and C closest to P), the proof adapts inscribed angle properties alongside exterior angle relations, using similar cross-chord connections as in the similar triangles proof. Draw chords AD and CB. Apply the law of sines in △PAD and △PCB. In △PAD, \frac{PA}{\sin \angle PDA} = \frac{PD}{\sin \angle PAD}; in △PCB, \frac{PC}{\sin \angle PBC} = \frac{PB}{\sin \angle PCB}. With ∠PDA = ∠PBC (subtending arc AC) and ∠PAD = ∠PCB (subtending arc BD), the ratios give \frac{PA}{PD} = \frac{PC}{PB}, or \frac{PA}{PC} = \frac{PD}{PB}, leading to PA \cdot PB = PC \cdot PD. This highlights the circle-specific angle measures tying the products equal.^[20] These angle-based derivations trace back to Euclid's foundational work on circle theorems in Elements Book III, where propositions on inscribed and central angles (e.g., Proposition 21) were established, later adapted by geometers like Apollonius to encompass power relations through trigonometric extensions in Renaissance texts.

Single Circle Properties

Radical Axis

The radical axis of two circles is the locus of all points that have equal power with respect to both circles.^[21] For two non-concentric circles with centers at (a_1, b_1) and (a_2, b_2), and radii r_1 and r_2, respectively, this condition yields the equation of a straight line given by

$2(a_1 - a_2)x + 2(b_1 - b_2)y = a_1^2 + b_1^2 - r_1^2 - (a_2^2 + b_2^2 - r_2^2),

which can be derived by subtracting the standard equations of the two circles.^[22] Geometrically, the radical axis is perpendicular to the line joining the centers of the two circles, and its position along this line of centers is determined by the radii and the distance d between the centers.^[21] Specifically, the intersection point divides the line segment between the centers such that the distance from the first center is d_1 = (d^2 + r_1^2 - r_2^2)/(2d).^[21] A key property is that if the two circles intersect at two points, the radical axis coincides with the common chord passing through those points; however, the radical axis is defined even for non-intersecting circles.^[22] For example, consider two non-intersecting circles; any point on their radical axis allows tangents of equal length to be drawn to both circles, reflecting the equal power at those points.^[23]

Orthogonal Circles

Two circles in the plane intersect orthogonally if they cross at right angles, meaning the tangents to the circles at each intersection point are perpendicular.^[24]^[25] This property implies that the line segment joining the centers of the two circles forms a right triangle with the radii drawn to an intersection point.^[25] The orthogonality condition can be expressed using the power of a point: for two circles with centers O_1 and O_2 and radii r_1 and r_2, the circles are orthogonal if the power of O_1 with respect to the second circle equals r_1^2, which simplifies to the distance between centers satisfying |O_1O_2|^2 = r_1^2 + r_2^2.^[24]^[4] Equivalently, if a circle centered at a point A outside another circle is orthogonal to it, the square of the radius of the first circle equals the power of A with respect to the second circle.^[4]^[25] To construct a circle orthogonal to a given circle c with center O and radius r, select a point A and its inverse point B with respect to c, then draw the circle passing through A and B; this new circle will intersect c orthogonally.^[26] The equation of such an orthogonal circle centered at (x_0, y_0) with respect to a circle (x - a)^2 + (y - b)^2 = r^2 is (x - x_0)^2 + (y - y_0)^2 = (x_0 - a)^2 + (y_0 - b)^2 - r^2, where the right-hand side is the power of (x_0, y_0) relative to the given circle.^[25] Orthogonal circles play a key role in inversive geometry, where inversion with respect to one circle maps the other circle to itself if they intersect orthogonally, thereby preserving angles between curves.^[24] This invariance facilitates the study of circle packings and conformal mappings in the plane.^[24]

Multiple Circles and Constructions

Common Power of Two Circles

In geometry, the common power of two circles with respect to a point P refers to the equal value of the power of P relative to each circle when P lies on their radical axis; for points off this axis, the powers differ, and the difference in powers defines the locus equation for the axis itself.^[1]^[23] The power of a point P with respect to a single circle with center O and radius r is given by PO^2 - r^2, and this extends naturally to multiple circles by comparing these values.^[4] The radical axis of two circles with centers O_1, O_2 and radii r_1, r_2 is the set of points P satisfying \pow_{\omega_1}(P) = \pow_{\omega_2}(P), or equivalently,

PO_1^2 - r_1^2 = PO_2^2 - r_2^2.

Rearranging yields the difference of powers:

\pow_{\omega_1}(P) - \pow_{\omega_2}(P) = 2 \mathbf{P} \cdot (\mathbf{O_2} - \mathbf{O_1}) + (|\mathbf{O_1}|^2 - r_1^2 - |\mathbf{O_2}|^2 + r_2^2),

a linear equation in the coordinates of P, confirming the radical axis is a straight line perpendicular to the line joining the centers.^[1]^[23] On this axis, the common power k = \pow_{\omega_1}(P) = \pow_{\omega_2}(P) varies with position along the line.^[4] For three circles, the radical axes of each pair intersect at the radical center, a point where the powers with respect to all three circles are equal, sharing a single common power value.^[1]^[23] This concurrency holds provided the circles are non-concentric, enabling constructions like the orthocenter via radical axes of circles on triangle sides.^[23] Key properties include that the difference in powers between the two circles remains constant along any line parallel to the radical axis, reflecting the linear nature of the difference equation projected onto directions perpendicular to the line of centers.^[1] Additionally, if the circles intersect orthogonally (as defined in the orthogonal circles section), the common power is zero at their intersection points, since each such point lies on both circles.^[4] Consider two overlapping circles: one centered at (0,0) with radius 2, the other at (3,0) with radius 2. Their radical axis is the line x = 1.5. At the point P = (1.5, 0) on this axis, the common power is

k = (1.5)^2 + 0^2 - 2^2 = 2.25 - 4 = -1.75,

verified similarly for the second circle: (1.5 - 3)^2 + 0^2 - 2^2 = 2.25 - 4 = -1.75. At points farther along the axis, such as P = (1.5, 1), k = 1.5^2 + 1^2 - 4 = 2.25 + 1 - 4 = -0.75, illustrating the variation.^[1]

Similarity Points

The similarity points of two circles, also known as the centers of similitude, are specific points from which tangents drawn to each circle have lengths proportional to the radii of the respective circles, rendering the circles homothetic with respect to that point.^[27] This property implies that the two circles appear similar when viewed from these points, with the ratio of similarity equal to the ratio of their radii.^[28] There are two such points: the external center of similitude, associated with direct (positive) similarity where the orientations of the circles align, and the internal center of similitude, associated with opposite (negative) similarity where the orientations are reversed.^[28] The external center is the intersection point of the two external common tangents to the circles, while the internal center is the intersection point of the two internal common tangents.^[29] These points lie on the line joining the centers of the two circles, denoted O_1 and O_2 with radii r_1 and r_2, respectively.^[28] Their positions are found by solving for points P where the signed distances d_1 = |PO_1| and d_2 = |PO_2| satisfy \frac{d_1}{r_1} = \pm \frac{d_2}{r_2}, with the positive sign for the external center and the negative sign for the internal center (accounting for direction along the line).^[27] This condition derives from equating the normalized powers of the point with respect to each circle, since the power equals the square of the tangent length and the proportionality t_1 / t_2 = r_1 / r_2 implies \frac{\text{power}_1}{r_1^2} = \frac{\text{power}_2}{r_2^2}, leading to \left( \frac{d_1}{r_1} \right)^2 = \left( \frac{d_2}{r_2} \right)^2 and thus the \pm solutions.^[27] To locate these points explicitly, consider the line of centers with distance d between O_1 and O_2. Placing O_1 at position 0 and O_2 at d, the external center P_e is at x = \frac{r_1 d}{r_1 - r_2} (assuming r_1 > r_2; otherwise adjust the denominator), dividing the segment externally in the ratio r_1 : r_2. The internal center P_i is at x = \frac{r_1 d}{r_1 + r_2}, dividing the segment internally in the same ratio.^[30]^[31] These positions can be obtained by solving the linear equations from the \pm condition, though the squared form yields a quadratic equation in x: \left( \frac{x}{r_1} \right)^2 = \left( \frac{x - d}{r_2} \right)^2, which expands to x^2 r_2^2 = (x - d)^2 r_1^2 and factors into the two solutions.^[27] Visually, the similarity points serve as origins for common tangents: from the external center, the non-crossing tangents touch both circles on the same side, while from the internal center, the crossing tangents touch on opposite sides, illustrating the direct and opposite similarities, respectively.^[29] This configuration highlights how the points unify the tangent structures of the two circles under homothety.^[27]

Tangent Circles to Two Given Circles

The problem of finding a circle tangent to two given circles constitutes a variant of Apollonius' problem, where the solutions form families rather than discrete points, determined by the relative positions and radii of the given circles. This setup arises in plane geometry when seeking circles that touch each of the two given circles at single points, with the type of tangency (external or internal) influencing the configuration.^[32] The power of the center of the tangent circle with respect to each given circle is adjusted by the tangency condition, leading to the center lying on the locus where the powers are related by the radius; in cases where the given circles have equal radii, this simplifies to the center having equal power with respect to both, placing it on their radical axis, with the radius then determining the precise distance from the given centers.^[33] The similarity points of the two given circles play a crucial role in this construction, as the centers of the tangent circles lie on lines connecting the similarity points to the centers of the given circles in special cases, and more generally, the similarity points serve as limiting points for the families of tangent circles, where the radius approaches zero.^[34] There are up to 8 tangent circles in discrete configurations (such as when a third condition like a point or line is implied in the variant), classified by combinations of external and internal tangency with each given circle: external-external, external-internal, internal-external, and internal-internal, with two solutions per type depending on the branch of the locus hyperbola or ellipse.^[35] To construct these tangent circles, one employs homothety centered at the external or internal similarity point, which maps one given circle to a circle tangent to the other; the ratio of the homothety is chosen based on the desired tangency type, ensuring the image circle touches the second given circle while preserving angles and ratios from the similarity center. For example, starting from the external similarity point S, a homothety with ratio k = (r ± r_2)/r_1 maps the first circle to a position where it is tangent to the second, with the sign depending on external or internal tangency. The process can be iterated or combined with inversion for complex cases, yielding the desired tangent circle.^[36]

Higher-Dimensional Extensions

Power with Respect to a Sphere

The power of a point P with respect to a sphere with center O and radius r is defined as \pi(P) = |PO|^2 - r^2.^[37] This quantity is independent of the coordinate system and reflects the relative position of P to the sphere: positive if P is outside the sphere, zero if on the surface, and negative if inside.^[38] Analogous to the planar case, if a line through P intersects the sphere at points A and B, the product of the directed distances PA \cdot PB equals the power \pi(P).^[39] A key theorem extends this to intersecting lines or secants in three dimensions, maintaining the segment product equality for any line through P. This property holds because the power is constant for all such lines, derived from the quadratic nature of the sphere's equation along the line. For two spheres with centers O_1, O_2 and radii r_1, r_2, the locus of points P with equal power \pi_1(P) = \pi_2(P) forms a plane known as the radical plane, perpendicular to the line joining O_1 and O_2.^[40] The equation of this plane can be obtained by subtracting the sphere equations, yielding a linear form.^[41] In three-dimensional inversive geometry, inversion with respect to a sphere preserves angles and maps spheres to spheres or planes, while the power of points transforms consistently under this mapping, enabling constructions like orthogonal spheres.^[42] For example, when two spheres intersect, their common chord is a circle lying in the radical plane, with the line of centers perpendicular to this plane.^[40] This configuration is fundamental in problems involving coaxial systems of spheres and radical centers.

Darboux Product

The Darboux product provides a generalization of the power of a point theorem to pairs of circles or spheres, measuring an invariant relation between them analogous to the power's role for a single circle and an external point. Introduced by Gaston Darboux in his 1872 memoir on relations among points, circles, and spheres, it is defined for two circles with centers A_1 and A_2 and radii r_1 and r_2 as the quantity |A_1 A_2|^2 - r_1^2 - r_2^2. This bilinear form extends naturally to spheres in three dimensions using the Euclidean distance between their centers.^[43] When one circle degenerates to a point (i.e., r_2 = 0), the Darboux product reduces to the classical power of that point with respect to the first circle, |A_1 A_2|^2 - r_1^2, recovering the quadratic expression from the circle's equation evaluated at the point. For non-degenerate circles, the product equals zero if and only if the circles are orthogonal, meaning their tangents intersect at right angles at intersection points. This relation holds similarly for spheres, where orthogonality means the spheres intersect such that their tangents are perpendicular at intersection points. When the circles intersect, the product equals $2 r_1 r_2 \cos \phi, where \phi is the angle between their tangents at the intersection point.^[44] The Darboux product is invariant under Möbius transformations, which preserve circles and spheres, making it a fundamental tool in inversive and conformal geometry. It plays a key role in enumerative geometry, such as determining conditions for circle configurations or solving problems like Apollonius' tangent circles, where the product constrains possible solutions. In the context of sphere power from the previous section, the Darboux product extends the unary power to binary interactions, facilitating analyses of coaxial systems and radical centers in higher dimensions.^[45] For a concrete example, consider two circles in the plane: one centered at the origin with radius 1, and another centered at (4,0) with radius 2. The distance between centers is 4, so the Darboux product is $4^2 - 1^2 - 2^2 = 16 - 1 - 4 = 11 > 0, indicating the circles are separate with no intersection. If the second center moves to (0.5,0), the product becomes $0.5^2 - 1^2 - 2^2 = 0.25 - 1 - 4 = -4.75 < 0, showing one circle inside the other without touching. These computations highlight the product's utility in classifying relative positions without solving intersection equations directly.^[43]

Advanced Theorems

Laguerre's Theorem

Laguerre's theorem generalizes the classical power of a point from circles to algebraic curves, particularly relevant for envelopes formed by pencils of circles. For a pencil of circles— a one-parameter family generated by linear combinations of two distinct circle equations—the envelope is the algebraic curve tangent to every circle in the pencil. The theorem states that the power of a point with respect to this envelope curve relates to tangency conditions: specifically, a point lies on the envelope (and thus is a point of tangency for some circle in the pencil) if and only if its power is zero. This power is defined as the value of the envelope's defining polynomial evaluated at the point, up to a scaling factor, ensuring consistency with the standard power for individual circles where the power equals the square of the tangent length from the point. In Laguerre geometry, the power serves as a fundamental metric, defining distances between points and circles as the square root of the absolute value of the power, which induces a non-Euclidean structure on the plane. The theorem links the power of a point relative to the envelope to geometric features such as caustics (envelopes of reflected rays from circle families) or evolutes (envelopes of normals), providing a measure of how the point interacts with the family's tangency loci. For instance, positive or negative power values indicate the side of the envelope on which the point lies, influencing properties like the number of real tangents from the point to circles in the pencil. Applications of Laguerre's theorem appear in differential geometry, where it aids analysis of circle families. Historically, the theorem was introduced by the French mathematician Edmond Laguerre (1834–1886) in the mid-19th century as part of his development of Laguerre geometry, a non-Euclidean framework emphasizing oriented circles and lines as primitives, with power-based distances enabling invariant treatments of tangency and envelopes.^[46] A representative example is the pencil of circles tangent to two fixed intersecting lines, whose centers lie along the angle bisectors and whose envelope consists of the two lines themselves (degenerate conic). The power of a point with respect to this envelope determines its position relative to the lines: zero power places it on one of the lines (tangency locus), while nonzero values measure signed distances influencing the configuration of tangent circles from that point.

Miquel's Theorem

Miquel's theorem, a key result in circle geometry, asserts that in a complete quadrilateral—formed by four lines in general position with six intersection points—the circumcircles of the four triangles created by these lines concur at a single point known as the Miquel point.^[47] Specifically, if the four lines intersect to form points A, B, C, D on the sides and diagonal points P and Q, then the circles passing through the triangles formed by pairs of these points (such as QAD, QBC, RAB, and RCD, where R is another intersection) all intersect at the Miquel point M.^[48] This concurrence ties directly to the power of a point: the Miquel point M has equal power with respect to each of the four circles (specifically, zero, as it lies on each). This equality underscores how the theorem leverages power invariance to ensure the circles share M as a common point beyond their defined triangles.^[47] A generalization of the theorem applies to triangles: given a triangle ABC with points X, Y, Z on sides BC, CA, AB respectively, the circumcircles of triangles AYZ, BZX, and CXY intersect at a single Miquel point.^[49] This pivot theorem variant extends the quadrilateral case by focusing on side points and circle intersections, maintaining the concurrence property. A proof sketch relies on radical axes: consider pairs of the four circles; their radical axes (loci of points with equal power to each pair) are the lines of the complete quadrilateral itself, which concur at the diagonal points. The radical center of any three circles lies on the fourth's radical axis with one of them, implying all four circles share a common point M where powers equalize, confirming concurrence.^[48] Named after French mathematician Auguste Miquel (1816–1851), the theorem was published in 1838 and builds on earlier projective ideas, including extensions related to Simson lines in triangle projections.^[50] For an illustrative example, envision a complete quadrilateral with four lines forming triangles TAB, SAC, UBD, and VCD (where T, S, U, V are intersection points); the circumcircles of these triangles intersect at M, the Miquel point, demonstrating the power equality as all powers from M to the circles are zero.^[47]

References

[1]
[PDF] Power of a Point - Yufei Zhao
Power of a point is a frequently used tool in Olympiad geometry. Theorem 1 (Power of a point). Let Γ be a circle, and P a point. Let a line through P meet Γ.
[2]
None
### Summary of Power of a Point from Ray Li's Lecture Notes
[3]
Jacob Steiner and the Power of a Point
Accordingly, Steiner precedes the definition of the power of a point with a geometrical locus having, ostensibly, nothing to do with circles—rather, a locus ...
[4]
Power of a Point
Definition: The power of A with respect to c = |OA|2 - r · pc = d2 - r · pc(A) = |OA|2 - r · Notation Note: Sved uses the notation P(c) for the power of the point ...Missing: mathematics | Show results with:mathematics
[5]
Power of a Point Theorem
The power of a point inside the circle is negative, whereas that of a point outside the circle is positive. This is exactly what one obtains from the algebraic ...
[6]
Power of a point: from Jakob Steiner to modern applications
Feb 24, 2022 · Power of a point: from Jakob Steiner to modern applications ... 1826, published posthumously by Fueter, R. and Gonseth, F. (1931) ...<|separator|>
[7]
[PDF] GEOMETRY REVISITED H. S. M. Coxeter S. L. Greitzer - Aproged
as the Steiner-Lehmus theorem. Papers on it appeared in various ... power of a point P with respect to a circk. dl - R, where d is the distance ...Missing: Jakob | Show results with:Jakob
[8]
[PDF] An Invariant Version of the Power of a Point Theorem - Naval Academy
Properties invariant under change of coordinates can be expressed as the vanishing of polynomials in brackets. P1, P2, P3 are collinear precisely when [123] = ...
[9]
None
Below is a merged summary of the connection between the Power of a Point and Inversive Geometry, consolidating all information from the provided segments into a single, detailed response. To maximize density and clarity, I will use a table in CSV format to organize key concepts, definitions, connections, interpretations, and references, followed by a narrative summary that ties everything together. This approach ensures all details are retained and easily accessible.
[10]
Euclid's Elements, Book III, Proposition 35 - Clark University
3. Then, since a straight line GF through the center cuts a straight line AC not through the center at right angles, it also bisects it, therefore AG equals GC.
[11]
Power of a Point Theorem - AoPS Wiki
### Summary of Intersecting Chords Case (Case 1) of Power of a Point Theorem
[12]
[PDF] Math 1312 Section 6.3 Line and Segment Relationships in the Circle
Example 10: 16 16 12 x Page 5 Theorem 5: If two chords intersect within a circle, then the product of the lengths of the segments (parts) of one chord is equal ...
[13]
Going Full Circle - Andrews University
30/d=tan 28° or d=30/tan 28°=56.4'. One can construct the center of the circle by locating the point of intersection of the bisectors of two chords of the ...
[14]
[PDF] Circle Definitions and Theorems
Intersecting Chords Lengths Thoerem If two chords intersect within a circle, then the product of the lengths of the segments of one chord is equal to the ...
[15]
How to Use the Secant-Secant Power Theorem - Dummies
are you sitting down — two secants!
[16]
Intersecting Secants Theorem - Varsity Tutors
The theorem applies only to secant segments from the same exterior point. · Use the formula M N ⋅ M O = M P ⋅ M Q to find unknown lengths. · Secants intersect the ...<|control11|><|separator|>
[17]
[PDF] Power of a Point - Yufei Zhao
Power of a point is a frequently used tool in Olympiad geometry. Theorem 1 (Power of a point). Let Γ be a circle, and P a point. Let a line through P meet Γ.Missing: secant- secant<|control11|><|separator|>
[18]
[PDF] Using Technology to Unify Geometric Theorems About the Power of ...
In the investigation, the students discovered and unified the four theorems associated with the power of a point: the secant-secant theorem, the secant-tangent ...
[19]
[PDF] ORMC AMC Group: Week 8 Geometry: Similar Triangles
Nov 20, 2022 · 2. Power of a Point. Show that in both diagrams below, PC · PD = PA · PB. (This will require 2 separate proofs).
[20]
Circle Geometry - Australian Mathematical Sciences Institute
The final theorems in this module combine similarity with circle geometry to produce three theorems about intersecting chords, intersecting secants, and the ...
[21]
Radical Line -- from Wolfram MathWorld
The radical line, also called the radical axis, is the locus of points of equal circle power with respect to two nonconcentric circles.
[22]
Radical axis - AoPS Wiki
### Summary of Radical Axis from AoPS Wiki
[23]
[PDF] Power of a Point and Radical Axis - NYC Math Team
§1 Power of a Point. This handout will cover the topic power of a point, and one of its more powerful uses in radical axes. Definition (Power of a Point).
[24]
Orthogonal Circles -- from Wolfram MathWorld
The orthogonal circle is the locus of a point whose polars with respect to the three given circles are concurrent (Lachlan 1893, p. 237). The following table ...
[25]
orthogonal circles - PlanetMath.org
Mar 22, 2013 · Two circles intersecting orthogonally (http://planetmath.org/ConvexAngle) are orthogonal curves and called orthogonal circles of each other.
[26]
[PDF] 1. Inversions.
Theorem. Any circle passing through a point A and its image. B under the inversion about a circle c is orthogonal to c. If B is not.
[27]
Circle of Similitude - Interactive Mathematics Miscellany and Puzzles
For the two points C1, C2 and the ratio R1/R2, this is exactly the circle of similitude defined above. Thus for this circle. d1/d2 = R1/R2. is automatic ...
[28]
Similitude Center -- from Wolfram MathWorld
If two similar figures lie in the plane but do not have parallel sides (i.e., they are similar but not homothetic), there exists a center of similitude, ...
[29]
Circle-Circle Tangents -- from Wolfram MathWorld
The point of intersection of the two crossing tangents is called the internal similitude center. The point of intersection of the extensions of the other two ...
[30]
External Similitude Center -- from Wolfram MathWorld
### Summary of Tangent Lengths or Ratios at the External Similitude Center
[31]
Internal Similitude Center -- from Wolfram MathWorld
The internal center of similitude of these two circles Si is the isogonal conjugate of the Gergonne point of DeltaABC . It is Kimberling center X_(55) and ...
[32]
[PDF] Apollonius problem
Apollonius in his treatise “contacts,” which was lost, formulated more generally the problem of finding circles which are tan‑ gent to three “things.
[33]
The locus of centre of circle tangent to two given circles
Jun 21, 2014 · What is the locus of the centre of circles that are tangent to two given circles? I had no idea how to approach the problem so I considered a ...Locus of centres of circles tangent to two fixed circles?Locus of the Centers of Circles. - Math Stack ExchangeMore results from math.stackexchange.com
[34]
http://users.math.uoc.gr/~pamfilos/eGallery/problems/Similarities.pdf
[35]
Apollonius' Problem -- from Wolfram MathWorld
Given three objects, each of which may be a point, line, or circle, draw a circle that is tangent to each. There are a total of ten cases.
[36]
Tangent Circles - Whistler Alley Mathematics
Sep 25, 2015 · The Gergonne construction: The dilation point or center of homothety of two circles is the point from which one of the circles may be dilated ...
[37]
Any open bounded subset of Rn has the same homotopy type as its ...
... power of a point with respect to a sphere is the difference of the squared distance to the sphere's center and the square of the radius. The radical plane ...
[38]
[PDF] Delaunay property and proximity results of the L-algorithm - HAL
Jul 11, 2022 · – Totality: the power of a point with respect to a sphere is negative (resp. zero or positive) if it is inside of (resp. on or outside of) ...
[39]
(PDF) On the Basic Properties and the Structure of Power Cells
May 15, 2024 · ... power of a point with respect to a sphere is the product of the distances from that. point to the two intersections with the sphere of an ...
[40]
[PDF] A treatise on the circle and the sphere
... define as the power of a point with regard to a circle the product of its oriented distances to any two points of the circle collinear with it (the square ...<|control11|><|separator|>
[41]
RadicalPlane - Maple Help
### Summary of Radical Plane and Power of a Point for Spheres
[42]
https://math.libretexts.org/Bookshelves/Geometry/Euclidean_Plane_and_its_Relatives_(Petrunin)/16%3A_Spherical_geometry/16.03%3A_Section_3-
[43]
XV. On systems of circles and spheres - Journals
... Darboux in several papers published during the years 1869‒1872. In Part I ... power of a point and a circle,” being due to Darboux, but the definition ...
[44]
II. On systems of circles and spheres | Proceedings of the Royal ...
Darboux was the first to give the definition of the power of two circles, in a paper “Sur les Relations entre les Groupes de Points, de Cercles, et de ...
[45]
On Systems of Circles and Spheres. [Abstract] - jstor
Darboux for the conception of the "power of two circles," or spheres,. -as an extension of Steiner's use of the "power of a point with respect to a circle ...
[46]
[PDF] arXiv:1802.05507v1 [math.DG] 15 Feb 2018
Feb 15, 2018 · This exposition gives an introduction to the theory of surfaces in Laguerre geometry and surveys some results, mostly obtained by the au- thors, ...
[47]
[PDF] Applications of Laguerre geometry in CAGD
Inserting relation (12) into the expression for c, we find that the locus of junction points c is a pe circle d. Application of the cyclographic map gives us ...
[48]
[PDF] Complete Quadrilaterals and the Miquel Point - Victor Rong
Jul 9, 2021 · Then the circumcircles of QAD, QBC, RAB, and RCD intersect at a point known as the Miquel point. M is the center of spiral similarity for pairs ...
[49]
[PDF] Geometry Unbound - Kiran S. Kedlaya
Mar 10, 2011 · what is the set of points with equal power with respect to the two circles? ... through a common point, called the Miquel point of the complete ...<|control11|><|separator|>
[50]
[PDF] Cyclic Quadrilaterals — The Big Picture - Yufei Zhao
Fact 1 (Miquel's Theorem). Let ABC be a triangle, and let X,Y,Z be points on lines BC, CA, AB, respectively. Assume that the ...
[51]
[PDF] Complete Quadrilaterals: Exploring the Elegance of Geometry
This statement is known as Miquel's theorem and the point M in question is referred to as Miquel's point. Proof: We prove the theorem using angles in the ...