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493. Ptolemy's Theorem

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. ...

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147. Ptolemy's Theorem

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. ...

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653. Ptolemy's Theorem

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*. ...

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654. Proof of Ptolemy's Theorem

1923
*
Mathematical Gazette
*

—By reversing the above steps, we can deduce Stewart’s

doi:10.2307/3604612
fatcat:lk2qdklperek7ed4ea77d5tsl4
*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. ...##
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Heron's Formula and Ptolemy's Theorem

2008
*
Formalized Mathematics
*

and the

doi:10.2478/v10037-008-0014-2
fatcat:irzbex4ztnamxoky2xo37r3fre
*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*...##
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Ptolemy's Theorem and Regular Polygons

1946
*
Mathematics Teacher
*

But without doubt,

doi:10.5951/mt.39.3.0117
fatcat:unpteebqjvdxfdtrvvpoedjoii
*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. ...##
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Converse of Ptolemy's Theorem on Stereographic Projection

1945
*
Proceedings of the National Academy of Sciences of the United States of America
*

Thus the only perspective conformalities upon a plane are

doi:10.1073/pnas.31.10.338
pmid:16588705
pmcid:PMC1078837
fatcat:gnwtc4hzwzgzte6prlz4626f4y
*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*. ...##
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Some Applications of Ptolemy's Theorem in Secondary School Mathematics

2020
*
European Journal of Pure and Applied Mathematics
*

We consider some applications of

doi:10.29020/nybg.ejpam.v1i1.3605
fatcat:rxj5jaboczb6pacjpyvzczw3zu
*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*. ...##
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An analogue of Ptolemy's theorem and its converse in hyperbolic geometry

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) . ...

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

1888
*
Nature
*

Further Use of

doi:10.1038/038149b0
fatcat:fu2gocy5z5gx5km5e7tqwucxga
*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. ...##
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PROOFS OF THE ADDITION AND SUBTRACTION FORMULAS BY MEANS OF PTOLEMY'S THEOREM

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. ...

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

2004
*
Annales Polonici Mathematici
*

We state and prove a

doi:10.4064/ap84-2-5
fatcat:n333ptyyivbrdnndn6qkrylqrm
*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. ...##
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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

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

fatcat:qvmltcbrv5gg7jkemsahnfuvym
*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 ...##
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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 ...##
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Circular reasoning

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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