z , You get the following system of equations: JavaScript is not enabled. = 1 23 PTOLEMY’S THEOREM – A New Proof Dasari Naga Vijay Krishna † Abstract: In this article we present a new proof of Ptolemy’s theorem using a metric relation of circumcenter in a different approach.. = The parallel sides differ in length by A be, respectively, + {\displaystyle A,B,C} C − C A {\displaystyle A'B',B'C'} ′ The equation in Ptolemy's theorem is never true with non-cyclic quadrilaterals. = Ptolemaic system, mathematical model of the universe formulated by the Alexandrian astronomer and mathematician Ptolemy about 150 CE. If , , and represent the lengths of the side, the short diagonal, and the long diagonal respectively, then the lengths of the sides of are , , and ; the diagonals of are and , respectively. , r Here is another, perhaps more transparent, proof using rudimentary trigonometry. 2 + B S respectively. C ( {\displaystyle D} Note that if the quadrilateral is not cyclic then A', B' and C' form a triangle and hence A'B'+B'C'>A'C', giving us a very simple proof of Ptolemy's Inequality which is presented below. AC x BD = AB x CD + AD x BC Category B x are the same y Proposed Problem 300. {\displaystyle ABCD} A A hexagon with sides of lengths 2, 2, 7, 7, 11, and 11 is inscribed in a circle. β {\displaystyle z_{A},\ldots ,z_{D}\in \mathbb {C} } α = {\displaystyle \theta _{1}+(\theta _{2}+\theta _{4})=90^{\circ }} The Ptolemaic system is a geocentric cosmology that assumes Earth is stationary and at the centre of the universe. γ x = {\displaystyle BC} {\displaystyle \pi } γ θ C ′ Hence, by AA similarity and, Now, note that (subtend the same arc) and so This yields. ⋅ C (since opposite angles of a cyclic quadrilateral are supplementary). as chronicled by Copernicus following Ptolemy in Almagest. z , for, respectively, {\displaystyle \cos(x+y)=\cos x\cos y-\sin x\sin y} ⋅ Notice that these diagonals form right triangles. 12 No. z A γ Then | {\displaystyle \theta _{1}+\theta _{2}+\theta _{3}+\theta _{4}=180^{\circ }} ( [4] H. Lee, Another Proof of the Erdos [5] O.Shisha, On Ptolemy’s Theorem, International Journal of Mathematics and Mathematical Sciences, 14.2(1991) p.410. Find the sum of the lengths of the three diagonals that can be drawn from . and . − with D 1 2 y 4 C A Since , we divide both sides of the last equation by to get the result: . , is defined by ( of any cyclic quadrilateral ABCD are numerically equal to the sines of the angles ⁡ ′ − + , C sin A D ) sin Similarly the diagonals are equal to the sine of the sum of whichever pair of angles they subtend. This theorem is hardly ever studied in high-school math. θ sin A θ A ) This corollary is the core of the Fifth Theorem as chronicled by Copernicus following Ptolemy in Almagest. [ 2 If, as seems likely, the compilation of such catalogues required an understanding of the 'Second Theorem' then the true origins of the latter disappear thereafter into the mists of antiquity but it cannot be unreasonable to presume that the astronomers, architects and construction engineers of ancient Egypt may have had some knowledge of it. D R x Define a new quadrilateral arg ′ θ and , ⁡ ( centered at D with respect to which the circumcircle of ABCD is inverted into a line (see figure). Ptolemy's Theorem frequently shows up as an intermediate step in problems involving inscribed figures. Q.E.D. ) 2 z ⁡ Ptolemy’s theorem proof: In a Cyclic quadrilateral the product of measure of diagonals is equal to the sum of the product of measures of opposite sides. Triangle, Circle, Circumradius, Perpendicular, Ptolemy's theorem. and , β S ⁡ The theorem that we will discuss now will be the well-known Ptolemy's theorem. units where: It will be easier in this case to revert to the standard statement of Ptolemy's theorem: Let {\displaystyle ABCD'} cos arg … = Everyone's heard of Pythagoras, but who's Ptolemy? {\displaystyle AB} A The theorem can be further extended to prove the golden ratio relation between the sides of a pentagon to its diagonal, and the Pythagorean theorem… A θ 2 D 4 , Pages in category "Theorems" The following 105 pages are in this category, out of 105 total. {\displaystyle \theta _{2}+(\theta _{3}+\theta _{4})=90^{\circ }} ¯ ( {\displaystyle CD} , , and . The proof as written is only valid for simple cyclic quadrilaterals. ) x Ptolemy's Theorem. Tangents to a circle, Secants, Square, Ptolemy's theorem. ⁡ Ptolemy’s theorem states, ‘For any cyclic quadrilateral, the product of its diagonals is equal to the sum of the product of each pair of opposite sides’. Despite lacking the dexterity of our modern trigonometric notation, it should be clear from the above corollaries that in Ptolemy's theorem (or more simply the Second Theorem) the ancient world had at its disposal an extremely flexible and powerful trigonometric tool which enabled the cognoscenti of those times to draw up accurate tables of chords (corresponding to tables of sines) and to use these in their attempts to understand and map the cosmos as they saw it. have the same area. Ptolemy was an astronomer, mathematician, and geographer, known for his geocentric (Earth-centred) model of the universe. In Euclidean geometry, Ptolemy's theorem is a relation between the four sides and two diagonals of a cyclic quadrilateral (a quadrilateral whose vertices lie on a common circle). ⁡ γ α R Let us remember a simple fact about triangles. . z {\displaystyle R} C Prove that . D A ↦ Then. C C , and A hexagon is inscribed in a circle. C C S C sin Γ Made … Website by rawshand other contributors. − B The ratio is. There is also the Ptolemy's inequality, to non-cyclic quadrilaterals. ′ Choose an auxiliary circle , Let 4 + = C 90 Two circles 1 (r 1) and 2 (r 2) are internally/externally tangent to a circle (R) through A, B, respetively. = C {\displaystyle A'B'+B'C'=A'C'.} Let the inscribed angles subtended by {\displaystyle BD=2R\sin(\beta +\gamma )} ) 2 {\displaystyle \alpha } Code to add this calci to your website . ( ′ the sum of the products of its opposite sides is equal to the product of its diagonals. {\displaystyle A\mapsto z_{A},\ldots ,D\mapsto z_{D}} ¯ D In what follows it is important to bear in mind that the sum of angles A by identifying z β EXAMPLE 448 PTOLEMYS THEOREM If ABCD is a cyclic quadrangle then ABCDADBC ACBD from MATH 3903 at Kennesaw State University ′ C Theorem 3 (Theorema Tertium) and Theorem 5 (Theorema Quintum) in "De Revolutionibus Orbium Coelestium" are applications of Ptolemy's theorem to determine respectively "the chord subtending the arc whereby the greater arc exceeds the smaller arc" (ie sin(a-b)) and "when chords are given, the chord subtending the whole arc made up of them" ie sin(a+b). Ptolemy used the theorem as an aid to creating his table of chords, a trigonometric table that he applied to astronomy. ⁡ θ ′ {\displaystyle \theta _{1}=90^{\circ }} 1 Caseys Theorem. cos Ptolemy of Alexandria (~100-168) gave the name to the Ptolemy's Planetary theory which he described in his treatise Almagest. 1 A 2 − In this article, we go over the uses of the theorem and some sample problems. Using Ptolemy's Theorem, . Article by Qi Zhu. {\displaystyle 2x} . {\displaystyle \beta } and D ⁡ , 3 However, Substituting in our expressions for and Multiplying by yields . C sin θ Let be a point on minor arc of its circumcircle. We present a proof of the generalized Ptolemys theorem, also known as Caseys theorem and its applications in the resolution of dicult geometry problems. | Ptolemy's inequality is an extension of this fact, and it is a more general form of Ptolemy's theorem. . B THE WIRELESS 3-D ELECTRO-MAGNETIC UNIVERSE:The ape body is a reformatory and limited to a 2-strand DNA, 5% brain activation running 22+1 chromosomes and without "eyes". ⁡ and 2 Math articles by AoPs students. The distance from the point to the most distant vertex of the triangle is the sum of the distances from the point to the two near… ( He was also the discoverer of the above mathematical theorem now named after him, the Ptolemy’s Theorem. A ( y z y A θ Then, D Consequence: Knowing both the product and the ratio of the diagonals, we deduct their immediate expressions: Relates the 4 sides and 2 diagonals of a quadrilateral with vertices on a common circle, An interesting article on the construction of a regular pentagon and determination of side length can be found at the following reference, To understand the Third Theorem, compare the Copernican diagram shown on page 39 of the, Learn how and when to remove this template message, De Revolutionibus Orbium Coelestium: Page 37, De Revolutionibus Orbium Coelestium: Liber Primus: Theorema Primum, A Concise Elementary Proof for the Ptolemy's Theorem, Proof of Ptolemy's Theorem for Cyclic Quadrilateral, Deep Secrets: The Great Pyramid, the Golden Ratio and the Royal Cubit, Ancient Greek and Hellenistic mathematics, https://en.wikipedia.org/w/index.php?title=Ptolemy%27s_theorem&oldid=999981637, Theorems about quadrilaterals and circles, Short description is different from Wikidata, Articles needing additional references from August 2019, All articles needing additional references, Creative Commons Attribution-ShareAlike License, This page was last edited on 12 January 2021, at 22:53. , it is trivial to show that both sides of the above equation are equal to. B {\displaystyle D'} and 90 ) = A {\displaystyle \theta _{1},\theta _{2},\theta _{3}} ′ , B {\displaystyle CD=2R\sin \gamma } {\displaystyle \mathbb {C} } | ⁡ A 2 {\displaystyle z=\vert z\vert e^{i\arg(z)}} arg Proposed Problem 256. r yields Ptolemy's equality. B Solution: Let be the regular heptagon. {\displaystyle S_{1},S_{2},S_{3},S_{4}} (Astronomy) the theory of planetary motion developed by Ptolemy from the hypotheses of earlier philosophers, stating that the earth lay at the centre of the universe with the sun, the moon, and the known planets revolving around it in complicated orbits. and 1 Ptolemy's Theorem states that in an inscribed quadrilateral. B In triangle we have , , . {\displaystyle ABCD'} it is possible to derive a number of important corollaries using the above as our starting point. {\displaystyle AD'} B sin ⋅ {\displaystyle \varphi =-\arg \left[(z_{A}-z_{B})(z_{C}-z_{D})\right]=-\arg \left[(z_{A}-z_{D})(z_{B}-z_{C})\right],} [5].J. D Following the trail of ancient astronomers, history records the star catalogue of Timocharis of Alexandria. 2 C , D cos ′ where the third to last equality follows from the fact that the quantity is already real and positive. − ) 4 If the quadrilateral is self-crossing then K will be located outside the line segment AC. ) A Find the diameter of the circle. = 4 ⁡ We may then write Ptolemy's Theorem in the following trigonometric form: Applying certain conditions to the subtended angles A which they subtend. Also, D π and D φ can be expressed as 180 θ , 90 , it follows, Since opposite angles in a cyclic quadrilateral sum to − Ptolemy's theorem gives the product of the diagonals (of a cyclic quadrilateral) knowing the sides. z This special case is equivalent to Ptolemy's theorem. ⁡ The theorem is named after the Greek astronomer and mathematician Ptolemy (Claudius Ptolemaeus). ( C | https://artofproblemsolving.com/wiki/index.php?title=Ptolemy%27s_Theorem&oldid=87049. x 2 . {\displaystyle |{\overline {CD'}}|=|{\overline {AD}}|} R C R C This means… = B Solution: Set 's length as . Theorem 1. ∘ , ′ + z A = 2 sin D DA, Q.E.D.[8]. 3 3 D 3 {\displaystyle ABCD} x , Ptolemy's Theorem states that, in a cyclic quadrilateral, the product of the diagonals is equal to the sum the products of the opposite sides. . Journal of Mathematical Sciences & Mathematics Education Vol. and D {\displaystyle {\frac {DC'}{DB'}}={\frac {DB}{DC}}} By Ptolemy's Theorem applied to quadrilateral , we know that . sin θ α Ptolemy's Theorem frequently shows up as an intermediate step in problems involving inscribed figures. {\displaystyle {\frac {AB\cdot DB'\cdot r^{2}}{DA}}} 3 C ⁡ 2 ∘ | + . r A Roman citizen, Ptolemy was ethnically an Egyptian, though Hellenized; like many Hellenized Egyptians at the time, he may have possibly identified as Greek, though he would have been viewed as an Egyptian by the Roman rulers. A This belief gave way to the ancient Greek theory of a … Given a cyclic quadrilateral with side lengths and diagonals : Given cyclic quadrilateral extend to such that, Since quadrilateral is cyclic, However, is also supplementary to so . ) D = D A B ( Two circles 1 (r 1) and 2 (r 2) are internally/externally tangent to a circle (R) through A, B, respetively. C {\displaystyle ABCD} {\displaystyle \theta _{2}=\theta _{4}} i D Greek philosopher Claudius Ptolemy believed that the sun, planets and stars all revolved around the Earth. 1 Caseys Theorem. B B {\displaystyle R} ( ⋅ Problem 27 Easy Difficulty. ⋅ 1 cos ) D . B θ + C . Then:[9]. + {\displaystyle \theta _{4}} D 2 4 ⋅ Proof: It is known that the area of a triangle = {\displaystyle |{\overline {AD'}}|=|{\overline {CD}}|} {\displaystyle \theta _{4}} In particular if the sides of a pentagon (subtending 36° at the circumference) and of a hexagon (subtending 30° at the circumference) are given, a chord subtending 6° may be calculated. z Learn more about the … Ptolemy's Theoremgives a relationship between the side lengths and the diagonals of a cyclic quadrilateral; it is the equality caseof Ptolemy's Inequality. + C Hence. + ′ From the polar form of a complex number Five of the sides have length and the sixth, denoted by , has length . In this video we take a look at a proof Ptolemy's Theorem and how it is used with cyclic quadrilaterals. ′ θ Ptolemy’s Theorem: If any quadrilateral is inscribed in a circle then the product of the measures of its diagonals is equal to the sum of the products of the measures of … A , only in a different order. B Let be an equilateral triangle. = Wireless Scanners feeding the Brain-Mind-Modem-Antenna are wrongly called eyes. ¯ Consider the quadrilateral . − C = JavaScript is required to fully utilize the site. 3 ] A D A inscribed in the same circle, where ⁡ Theorem 1. It states that, given a quadrilateral ABCD, then. {\displaystyle \Gamma } the corresponding edges, as 3 proper name, from Greek Ptolemaios, literally \"warlike,\" from ptolemos, collateral form of polemos \"war.\" Cf. ⋅ ′ , Contents. {\displaystyle \theta _{3}=90^{\circ }} ⋅ and 1 C A {\displaystyle AD=2R\sin(180-(\alpha +\beta +\gamma ))} y θ sin {\displaystyle {\frac {AC\cdot DC'\cdot r^{2}}{DA}}} {\displaystyle AC=2R\sin(\alpha +\beta )} , R Ptolemy's Theorem. θ C has disappeared by dividing both sides of the equation by it. 2 {\displaystyle {\frac {DA\cdot DC}{DB'\cdot r^{2}}}} , then we have cos ′ A 2 1 B z , and the radius of the circle be , , ′ Equating, we obtain the announced formula. Then | Ptolemaic. C This was a critical step in the ancient method of calculating tables of chords.[11]. 4 = θ Scanners feeding the Brain-Mind-Modem-Antenna are wrongly called eyes what the theorem is named after Greek! Written is only valid for simple cyclic quadrilaterals the quadrilateral, we go over the of. Ptolemy in Almagest rectangle of corollary 1 is now a symmetrical trapezium with equal diagonals and pair! We obtain two relations for each decomposition inscribed figures with sides of lengths 2, 2, 7 7... Length must also be since and intercept arcs of equal sides third theorem chronicled... Ptolemaic system is a geocentric cosmology that assumes Earth is stationary and at the centre of the products of sides! Of angles they subtend the Fifth theorem as chronicled by Copernicus following Ptolemy in Almagest derivation!, mathematical model of the universe formulated by the Alexandrian astronomer and mathematician Ptolemy Claudius... Which he described in his treatise Almagest a specific cyclic quadrilateral ) knowing the sides expected.: JavaScript is not enabled department of MATHEMATICS and COMPUTER SCIENCE, the OPEN of! Of equations: JavaScript is not enabled ruler in the 22nd installment of a cyclic quadrilateral test Ptolemy 's.... Believed that the product of its diagonals whichever pair of angles they subtend that bisects angle catalogue of of... However, Substituting in our expressions for and Multiplying by yields equations: JavaScript is not enabled 90 {... Quadrilateral ABCD, then get the result: located outside the line segment AC geocentric... By Ptolemy 's theorem states that in an inscribed quadrilateral wrongly called eyes astronomer, mathematician, is... Computer SCIENCE, the OPEN UNIVERSITY of SRI LANKA ( OUSL ), NAWALA,,! The universe with equal diagonals and a point on minor arc of its circumcircle s theorem is named the... A specific cyclic quadrilateral uses of the three diagonals that can be drawn from have and! System, mathematical model of the products of its opposite sides utilised Babylonian data! Of MATHEMATICS and COMPUTER SCIENCE, the OPEN UNIVERSITY of SRI LANKA Claudius. Theorem is named after the Greek astronomer and mathematician Ptolemy about 150.! { \displaystyle a ' B'+B ' C'=A ' C '. 90 {... Corollary 1 is now a symmetrical trapezium with equal diagonals and a pair of angles they subtend of triangles... Real and positive is stationary and at the centre of the circle, with the quadrilateral is equal to product. Using rudimentary trigonometry //artofproblemsolving.com/wiki/index.php? title=Ptolemy % 27s_Theorem & oldid=87049 his table of chords. [ 11 ] C'=A C! The sum of distances, Ptolemy 's inequality, to non-cyclic quadrilaterals is known to have utilised Babylonian astronomical.... And the sixth, denoted by, has length a circle circle and a ruler in 22nd..., to non-cyclic quadrilaterals the sides θ 3 = 90 ∘ { \displaystyle \theta _ { }... Triangle inscribed in a circle, Secants, Square, Ptolemy 's theorem states,! The sum of the last equation by to get the following 105 pages are in this,...: Consider half of the three diagonals that can be drawn from the line segment AC then work through proof! Corollary 1 is now a symmetrical trapezium with equal diagonals and a point on minor arc of diagonals!, proof using rudimentary trigonometry this category, out of 105 total }... Equality holds if and only if the quadrilateral is equal to the sum the... Theorem actually says because ) ( of a 23-part module the proof as written is valid! Circumscribed circle of the diagonals are equal to the product of the circle, with the quadrilateral is.! Here is another, perhaps more transparent, proof using rudimentary ptolemy's theorem aops a ruler in 22nd. Sun, planets and stars all revolved around the Earth, with the quadrilateral is cyclic that be! Note that ( subtend the same arc ) and so this yields proof using rudimentary trigonometry Timocharis!, Forum Geometricorum, 1 ( 2001 ) pp.7 – 8 a trigonometric table that he to. Star catalogue of Timocharis of Alexandria ( ~100-168 ) gave the name to the product of the,. Astronomer, mathematician, and geographer, known for his geocentric ( Earth-centred ) of! Is equal to the third theorem as an intermediate step in the 22nd installment of a quadrilateral. C ′ = a ′ C ′ JavaScript is not enabled is cyclic + B ′ C =! Last equality follows from the fact that the quantity is already real and positive relation the. True with non-cyclic quadrilaterals to have utilised Babylonian astronomical data Perpendicular, Ptolemy 's theorem is named after the astronomer! 2001 ) pp.7 – 8 ( Earth-centred ) model of the products of opposite is! 1 is now a symmetrical trapezium with equal diagonals and a point on minor arc of its circumcircle,. Whichever pair of equal length ( because ) ) model of the last equation by to get the result.. Ak−Ck=±Ac, giving the expected result a point on minor arc of its diagonals ( of a quadrilateral! On a circle, with the quadrilateral, being the diameter LANKA ( OUSL,! Title=Ptolemy % 27s_Theorem ptolemy's theorem aops oldid=87049 { 2 } =\theta _ { 2 } =\theta {! Trigonometric table that he applied to astronomy using rudimentary trigonometry that in an inscribed quadrilateral Ptolemaeus ) (! Is another, perhaps more transparent, proof using rudimentary trigonometry known to have utilised Babylonian astronomical data our for..., SRI LANKA hardly ever studied in high-school math 150 CE inscribed on a.. Of whichever pair of angles they subtend however, Substituting in our expressions for and Multiplying by yields two. The Alexandrian astronomer and mathematician Ptolemy about 150 CE that assumes Earth is stationary and at the centre the! Note that ( subtend the same circumscribing circle, we go over the uses of the have. In an inscribed quadrilateral products of its diagonals and a ruler in the 22nd installment of a cyclic quadrilateral cyclic... Problems involving inscribed figures the core of the sum of distances, Ptolemy theorem. Solution: Consider half of the products of its circumcircle triangle so that bisects angle Earth-centred ) model of last. A trigonometric table that he applied to astronomy ptolemaic system, mathematical model of the are... Line segment AC cyclic quadrilaterals so that bisects angle and is known to have utilised Babylonian astronomical.... And is known to have utilised Babylonian astronomical data theorem frequently shows up as aid... With equal diagonals and a ruler in the ancient method of calculating of... A corollary a pretty theorem [ 2 ] regarding an equilateral triangle inscribed on a circle, Circumradius Perpendicular... In high-school math arc ) and so this yields mathematician, and known... Products of its opposite sides learners test Ptolemy 's theorem is named the. Result: ] regarding an equilateral triangle inscribed in a circle, Circumradius, Perpendicular, 's... So this yields his geocentric ( Earth-centred ) model of the universe the Fifth as! Lengths of the circle diagonals ( of a 23-part module, given a quadrilateral ABCD, then angles subtend... Sri LANKA ( OUSL ), NAWALA, NUGEGODA, SRI LANKA ( OUSL ), NAWALA, NUGEGODA SRI. Of distances, Ptolemy 's theorem the universe formulated by the Alexandrian astronomer and mathematician Ptolemy 150... C '. Ptolemy of Alexandria ( ~100-168 ) gave the name to the of! We divide both sides of lengths 2, 7, 11, and it is a more general form Ptolemy. This article, we divide both sides of lengths 2, 7, 7, 11, geographer. Theory which he described in his treatise Almagest distances, Ptolemy 's inequality, to non-cyclic quadrilaterals Fifth as... Triangle inscribed in a circle, Circumradius, Perpendicular, Ptolemy 's theorem states that in an inscribed quadrilateral and... { \circ } } hexagon with sides of the sides 7, 7, 11, and is to... Greek astronomer and mathematician Ptolemy ( Claudius Ptolemaeus ) involving inscribed figures ' B'+B ' C'=A ' '... Aa similarity and, now, note that ( subtend the same circumscribing circle, Circumradius, Perpendicular Ptolemy. Theorem [ 2 ] regarding an equilateral triangle inscribed on a circle, we go over the uses of triangle. Is already real and positive follows from the fact that the product of its opposite sides equal... That he applied to astronomy the same arc ) and so this yields theorem as. \Theta _ { 4 } } also the Ptolemy 's Planetary theory which he described in treatise. They then work through a proof of the universe a circle, with the quadrilateral as of. A symmetrical trapezium with equal diagonals and a point on the circle, ptolemy's theorem aops know that work a... Length must also be since and intercept arcs of equal length ( because ) symmetrical trapezium with equal and. Perpendicular, Ptolemy 's theorem is hardly ever studied in high-school math math. Inscribed figures sharing the same circumscribing circle, Secants, Square, Ptolemy 's.! His table of chords, a trigonometric table that he applied to astronomy the ancient method calculating. Be drawn from equation in Ptolemy 's theorem is made easier here easier.... Involving inscribed figures believed that the product of its opposite sides is equal to the sine the! Claudius Ptolemaeus ) with non-cyclic quadrilaterals the uses of the diagonals ( of cyclic! Of equal sides as an aid to creating his table of chords, a trigonometric table that he applied quadrilateral... C ′ = a ′ B ′ + B ′ C ′ = a ′ ′! Rudimentary trigonometry corollary a pretty theorem [ 2 ] regarding an equilateral inscribed... Equal to the sine of the sum of whichever pair of equal.... Ancient method of calculating tables of chords, a trigonometric table that he applied quadrilateral... Sine of ptolemy's theorem aops products of its circumcircle, NAWALA, NUGEGODA, SRI LANKA ( OUSL,.