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### 493. Ptolemy's Theorem

P. J. Harding
1916 Mathematical Gazette
Ptolemy's Theorem. London, 1832-40. P. J. HARDING. 494. [K1. 6. a.]  ...  Ptolemy's Theorem. The Philosophical Magazine, Comprehending the various branches of science, the liberal and fine arts (geology), agriculture, manufactures, and commerce. ByA. Tilloch.  ...

### 147. Ptolemy's Theorem

A. G. Sillitto
1966 Mathematical Gazette
Ptolemy's Theorem. The important consequence of the cyclically lies in the proportionalities which are written into the diagram.  ...  It is in fact the generalised Apollonius' Theorem (see eg.  ...

### 653. Ptolemy's Theorem

A. A. Krishnaswami Ayyangar
1923 Mathematical Gazette
Ptolemy’s Theorem. The following two proofs of Ptolemy’s theorem are new and interesting.  ...  Of them, one does not involve the principle of ratios and similar triangles, while the other shows the connection of Ptolemy’s theorem with Stewart’s Theorem.  ...

### 654. Proof of Ptolemy's Theorem

E. M. Langley
1923 Mathematical Gazette
—By reversing the above steps, we can deduce Stewart’s theorem from Ptolemy’s theorem. Mysore, India. A. A. KrisHNaswaMI AyYANGAR, M.A., L.T. 654. [K!.8.b.] Proof of Ptolemy’s Theorem.  ...  Applying Stewart’s Theorem to the triangle A BD, we get XD. AB + BX . AD® =BD(AA*+- BA . AD). cecccccocsececcosses (I) From the properties of the cyclic quadrilateral, we have (i) BX. XD=AX.  ...

### Heron's Formula and Ptolemy's Theorem

Marco Riccardi
2008 Formalized Mathematics
and the Ptolemy's theorem.  ...  These are elementary theorems included in every handbook of Euclidean geometry and trigonometry: the law of cosines, the Heron's formula, the isosceles triangle theorem, the intersecting chords theorem  ...

### Ptolemy's Theorem and Regular Polygons

L. S. Shively
1946 Mathematics Teacher
But without doubt, Ptolemy's theorem concerning an inscribed quadrilateral is deserving of a high place.  ...  If this were done, there is little doubt that the theorem of Pythagoras would in the opinion of most people, head the list. It is not so clear just how other important theorems would be ranked.  ...  But without doubt, Ptolemy’s theorem concerning an inscribed quadrilateral is deserving of a high place.  ...

### Converse of Ptolemy's Theorem on Stereographic Projection

E. Kasner, J. De Cicco
1945 Proceedings of the National Academy of Sciences of the United States of America
Thus the only perspective conformalities upon a plane are Ptolemy's stereographic projection of the sphere (or the limiting case of the plane). 2. Beginning of the Proof of the Theorem.  ...  Our result may be stated as follows: THEOREM.  ...

### Some Applications of Ptolemy's Theorem in Secondary School Mathematics

Samed Jahangir Aliyev, Shalala A. Hamidova, Goncha Z. Abdullayeva
2020 European Journal of Pure and Applied Mathematics
We consider some applications of Ptolemy's theorem. In particular, we nd a criterion for constructing an inscribed hexagon.  ...  Though, this is exactly what Ptolemy's famous theorem is about.  ...  In this work, we consider some applications of Ptolemy's theorem.  ...

### An analogue of Ptolemy's theorem and its converse in hyperbolic geometry

Joseph Valentine
1970 Pacific Journal of Mathematics
THEOREM. Four points P lf P 2 , P 3 , P 4 of the hyperbolic plane lie on a circle, line, horocycle, or one branch of an equidistant curve if and only if the determinant 817  ...  The following theorem, which is a hyperbolic analogue of Ptolemy's Theorem and its converse, has now been obtained. THEOREM 3.5 .  ...  It will be recalled that Ptolemy's Theorem and its converse for the euclidean plane may be stated as follows. THEOREM (Ptolemy) .  ...

### Further Use of Ptolemy's Theorem (Euclid, VI. D.) for a Problem in Maxima and Minima

E. M. LANGLEY
1888 Nature
Further Use of Ptolemy's Theorem (Euclid, VI. D.) for a Problem in Maxima and Minima. To find E within Ll.ABC such that AE sin BEC + BE sin CEA + CE sin AEB shall be a maximum.  ...

### PROOFS OF THE ADDITION AND SUBTRACTION FORMULAS BY MEANS OF PTOLEMY'S THEOREM

Albert Babbitt
1917 School Science and Mathematics
theorem and the Law of Sines, the addition and subtraction formulas of trigonometry can be easily derived.  ...  Ptolemy^s theorem states that^the rectangle contained by the diagonals of a quadrilateral figure inscribed in a circle is equal to both the rectangles contained by its opposite sides.'91 By means of this  ...  "It is believed that the elegant theorem generally known as Ptolemy's Theorem is due to Hipparchus and was copied from him by Ptolemy." Ball, A Short History of Mathematics (1909), p. 88.  ...

### A pseudo-trigonometry related to Ptolemy's theorem and the hyperbolic geometry of punctured spheres

Joachim A. Hempel
2004 Annales Polonici Mathematici
We state and prove a theorem resembling Ptolemy's classical theorem on cyclic quadrilaterals and three general lemmas on intersections of shortest (in the sense of pseudo-length) geodesic joins.  ...  A theorem like that of Ptolemy. The next theorem bears an uncanny similarity to Ptolemy's famous theorem on cyclic quadrilaterals.  ...  However we give another proof, which more clearly exhibits the similarity to Ptolemy's Theorem. Proof.  ...

### International Journal of Statistics and Applied Mathematics 2018; 3(1): 98-100 The proof of trigonometric formula for the sum and difference of the two angles by Ptolemy's theorem

Xintong Yang, Xintong Correspondence, Yang, Xintong Yang
2018 Maths   unpublished
Based on the proof of the sine formula of two angles provided by former researchers, we gave the proofs of they other three formulas with the Ptolemy's theorem.  ...  The proofs with Ptolemy's theorem about other three formulas have not been found yet. In fact, they could also be proved with the Ptolemy's theorem.  ...  CP  is the diameter of the circle O too,     90 CBP , ) sin( sin sin          BDC BPC BC Then we can know form the Ptolemy's theorem: AD BC CD AB AC BD      , namely ) sin( cos sin sin  ...

### Page 784 of School Science and Mathematics Vol. 17, Issue 9 [page]

1917 School Science and Mathematics
From Ptolemy’s theorem it follows that ef = ac+hd. Dividing through by f?  ...  , we get e ac bd = 4 Ol f pf Pp € acobd os 4. (1) ee oe ee 1Cajori, A History of Mathematics (1909), p. 57 “It is believed that the elegant theorem generally known as Ptolemy’s Theorem is due to Hipparchus  ...

### Circular reasoning

A. Glassner
1998 IEEE Computer Graphics and Applications
Ptolemy's Theorem Our second topic this month is Ptolemy's Theorem. Claudius Ptolemy (AD ~87~150) was an astronomer, mathematician, and geographer who lived in Alexandria, Greece.  ...  To prove Ptolemy's Theorem, we'll create an additional point K on line DB, such that ∠KCD = ∠BCA, as in Figure 3 .  ...
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