In mathematics, **Casey's theorem**, also known as the generalized Ptolemy's theorem, is a theorem in Euclidean geometry named after the Irish mathematician John Casey.

Let be a circle of radius . Let be (in that order) four non-intersecting circles that lie inside and tangent to it. Denote by the length of the exterior common bitangent of the circles . Then:^{ [1] }

Note that in the degenerate case, where all four circles reduce to points, this is exactly Ptolemy's theorem.

The following proof is attributable^{ [2] } to Zacharias.^{ [3] } Denote the radius of circle by and its tangency point with the circle by . We will use the notation for the centers of the circles. Note that from Pythagorean theorem,

We will try to express this length in terms of the points . By the law of cosines in triangle ,

Since the circles tangent to each other:

Let be a point on the circle . According to the law of sines in triangle :

Therefore,

and substituting these in the formula above:

And finally, the length we seek is

We can now evaluate the left hand side, with the help of the original Ptolemy's theorem applied to the inscribed quadrilateral :

It can be seen that the four circles need not lie inside the big circle. In fact, they may be tangent to it from the outside as well. In that case, the following change should be made:^{ [4] }

If are both tangent from the same side of (both in or both out), is the length of the exterior common tangent.

If are tangent from different sides of (one in and one out), is the length of the interior common tangent.

The converse of Casey's theorem is also true.^{ [4] } That is, if equality holds, the circles are tangent to a common circle.

Casey's theorem and its converse can be used to prove a variety of statements in Euclidean geometry. For example, the shortest known proof^{ [1] }^{:411} of Feuerbach's theorem uses the converse theorem.

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In mathematics, a **parabola** is a plane curve which is mirror-symmetrical and is approximately U-shaped. It fits several other superficially different mathematical descriptions, which can all be proved to define exactly the same curves.

In mathematics, the **trigonometric functions** are real functions which relate an angle of a right-angled triangle to ratios of two side lengths. They are widely used in all sciences that are related to geometry, such as navigation, solid mechanics, celestial mechanics, geodesy, and many others. They are among the simplest periodic functions, and as such are also widely used for studying periodic phenomena, through Fourier analysis.

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A **right triangle** or **right-angled triangle** is a triangle in which one angle is a right angle. The relation between the sides and angles of a right triangle is the basis for trigonometry.

In geometry, a **solid angle** is a measure of the amount of the field of view from some particular point that a given object covers. That is, it is a measure of how large the object appears to an observer looking from that point. The point from which the object is viewed is called the *apex* of the solid angle, and the object is said to *subtend* its solid angle from that point.

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In Euclidean geometry, **Ptolemy's theorem** is a relation between the four sides and two diagonals of a cyclic quadrilateral. The theorem is named after the Greek astronomer and mathematician Ptolemy. Ptolemy used the theorem as an aid to creating his table of chords, a trigonometric table that he applied to astronomy.

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In trigonometry, the **law of cosines** relates the lengths of the sides of a triangle to the cosine of one of its angles. Using notation as in Fig. 1, the law of cosines states

In Euclidean plane geometry, a **tangent line to a circle** is a line that touches the circle at exactly one point, never entering the circle's interior. Tangent lines to circles form the subject of several theorems, and play an important role in many geometrical constructions and proofs. Since the tangent line to a circle at a point **P** is perpendicular to the radius to that point, theorems involving tangent lines often involve radial lines and orthogonal circles.

In geometry, a **pentagon** is any five-sided polygon or 5-gon. The sum of the internal angles in a simple pentagon is 540°.

In Euclidean geometry, a **bicentric quadrilateral** is a convex quadrilateral that has both an incircle and a circumcircle. The radii and center of these circles are called *inradius* and *circumradius*, and *incenter* and *circumcenter* respectively. From the definition it follows that bicentric quadrilaterals have all the properties of both tangential quadrilaterals and cyclic quadrilaterals. Other names for these quadrilaterals are **chord-tangent quadrilateral** and **inscribed and circumscribed quadrilateral**. It has also rarely been called a *double circle quadrilateral* and *double scribed quadrilateral*.

- 1 2 Casey, J. (1866). "On the Equations and Properties: (1) of the System of Circles Touching Three Circles in a Plane; (2) of the System of Spheres Touching Four Spheres in Space; (3) of the System of Circles Touching Three Circles on a Sphere; (4) of the System of Conics Inscribed to a Conic, and Touching Three Inscribed Conics in a Plane".
*Proceedings of the Royal Irish Academy*.**9**: 396–423. JSTOR 20488927. - ↑ Bottema, O. (1944).
*Hoofdstukken uit de Elementaire Meetkunde*. (translation by Reinie Erné as Topics in Elementary Geometry, Springer 2008, of the second extended edition published by Epsilon-Uitgaven 1987). - ↑ Zacharias, M. (1942). "Der Caseysche Satz".
*Jahresbericht der Deutschen Mathematiker-Vereinigung*.**52**: 79–89. - 1 2 Johnson, Roger A. (1929).
*Modern Geometry*. Houghton Mifflin, Boston (republished facsimile by Dover 1960, 2007 as Advanced Euclidean Geometry).

Wikimedia Commons has media related to . Casey's theorem |

- Weisstein, Eric W. "Casey's theorem".
*MathWorld*. - Shailesh Shirali:
*On a generalized Ptolemy Theorem*^{[ permanent dead link ]} - Crux Mathematicorum volume 22 issue 2 (it contains the article above)

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