Ptolemy's Theorem Research Articles

A Short Proof of a Theorem of Erdos and Mordell

In 1935, Paul Erdos conjectured that for any point I inside (or on the boundary of) a triangle ABC, the sum of the distances from I to the vertices is at least twice the sum of the distances from I to the sides of AABC. He further conjectured that equality would hold if and only if AABC is equilateral and I is its circumcenter. Though this is easy to state and understand, the first proof was discovered only in 1937, by L. J. Mordell. It is by no means an elementary one. The first elementary proof was found by D. K. Kazarinoff in 1945 (see his son's book [2]). It is so tricky that it seems artificial. The purpose of this note is to give a proof which seems natural and is accessible to college students. We need two preliminary results. First is the elementary fact that r + r1 2 for every r > 0, with equality if and only if r = 1 (to see this, expand the left side of (r 1)2 * r-> > 0). The second result is known as Ptolemy's theorem: Let ABCD be a convex quadrilateral inscribed in a circle. Then the sum of the products of the opposite sides is equal to the product of the diagonals:

Proof without Words: The law of cosines via Ptolemy's Theorem

(1992). Proof without Words: The law of cosines via Ptolemy's Theorem. Mathematics Magazine: Vol. 65, No. 2, pp. 103-103.

Proof without Words: The Law of Cosines via Ptolemy's Theorem

Proof without Words: The Law of Cosines via Ptolemy's Theorem

On Ptolemy′s theorem

On Ptolemy′s theorem

74.32 The golden ratio via Ptolemy's theorem

74.32 The golden ratio via Ptolemy's theorem

The number of intersections of random chords to a circle — third and fourth moments

In this paper the distributions of the number of intersections of three, four and five random chords to a circle are obtained by a reduction technique employing Ptolemy's theorem. These results are then used to obtain the skewness and kurtosis of the number of intersections of n random chords to a circle.

The number of intersections of random chords to a circle — third and fourth moments

In this paper the distributions of the number of intersections of three, four and five random chords to a circle are obtained by a reduction technique employing Ptolemy's theorem. These results are then used to obtain the skewness and kurtosis of the number of intersections of n random chords to a circle.

An Analogue of Ptolemy's Theorem in Spherical Geometry

(pl, P2, . . . X Pn+2) = COS(pip1/P) where i, j=1, 2, * , n+2, and Pi, P2, P * p+2 are n+2 points of the ndimensional spherical Sntp (the surface of a sphere of radius p in euclidean (n+1)-dimensional space, with shorter arc metric, where PPj-=Pipi denotes the distance of the points pi, pj). Haantjes [4] gave a proof of the spherical analogue of the ptolemaic inequality, and [5 ] developed techniques which give a new proof of Ptolemy's Theorem and its converse in the euclidean plane. It is further stated [5] that these techniques give proofs of the analogue of Ptolemy's Theorem and its converse in the spherical and hyperbolic planes. However, his analogue of the converse of Ptolemy's Theorem in the hyperbolic plane is false; e.g., the determinant Isinh2 ppj1/21 =0, where i, j= 1, 2, 3, 4 for four points on a horocycle in the hyperbolic plane (see [6]), and consequently the result for the spherical plane is not immediate. The purpose of this note is to give the geometrical significance of the determinant

An Analogue of Ptolemy's Theorem in Spherical Geometry

An Analogue of Ptolemy's Theorem in Spherical Geometry

147. Ptolemy's Theorem

147. Ptolemy's Theorem

53. Ptolemy's Theorem by Area Formulae

53. Ptolemy's Theorem by Area Formulae

Ptolemy's Theorem and Regular Polygons

It might be both profitable and interesting to attempt to evaluate the relative importance of some of the outstanding theorems of elementary geometry by means of a poll of mathematicians. 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. This judgemnt is warranted, both from the point of view of the use which Ptolemy made of it, and from a consideration of its power as a tool for geometric study.

Converse of Ptolemy's Theorem on Stereographic Projection

Converse of Ptolemy's Theorem on Stereographic Projection

Ptolemy's Theorem and certain Trigonometrical Formulæ

Let XOX′ = 1 be the diameter of a circle,Then the lines representing the sines and cosines of the angles,are marked in the figure.

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

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