, and Everyone's heard of Pythagoras, but who's Ptolemy? D D and 2 , for, respectively, γ , {\displaystyle D} ∘ Code to add this calci to your website . ⁡ R x Writing the area of the quadrilateral as sum of two triangles sharing the same circumscribing circle, we obtain two relations for each decomposition. ( D That is, C B 2 ) ∈ {\displaystyle AB} sin B {\displaystyle \theta _{4}} e 3 B , ( ⋅ , which they subtend. where equality holds if and only if the quadrilateral is cyclic. A Here is another, perhaps more transparent, proof using rudimentary trigonometry. z z 1 ⋅ ⁡ , B Ptolemy's Theorem states that in an inscribed quadrilateral. He lived in Egypt, wrote in Ancient Greek, and is known to have utilised Babylonian astronomical data. − 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). The Ptolemaic system is a geocentric cosmology that assumes Earth is stationary and at the centre of the universe. {\displaystyle \theta _{3}=90^{\circ }} + Construct diagonals and . B B | S Since , we divide both sides of the last equation by to get the result: . B B 2 {\displaystyle ABCD} = r C {\displaystyle \theta _{1}+\theta _{2}=\theta _{3}+\theta _{4}=90^{\circ }} A , {\displaystyle ABCD} ¯ = A C So we will need to recall what the theorem actually says. + Ptolemy's theorem gives the product of the diagonals (of a cyclic quadrilateral) knowing the sides. 4 {\displaystyle ABCD'} C {\displaystyle ABCD'} 4 Learners test Ptolemy's Theorem using a specific cyclic quadrilateral and a ruler in the 22nd installment of a 23-part module. z C {\displaystyle AD=2R\sin(180-(\alpha +\beta +\gamma ))} Ptolemy’s Theorem”, Global J ournal of Advanced Research on Classical and Modern Geometries, Vol.2, I ssue 1, pp.20-25, 2013. C , as chronicled by Copernicus following Ptolemy in Almagest. 4 A {\displaystyle \theta _{1}+\theta _{2}+\theta _{3}+\theta _{4}=180^{\circ }} γ , Hence, by AA similarity and, Now, note that (subtend the same arc) and so This yields. The parallel sides differ in length by ] D = C Following the trail of ancient astronomers, history records the star catalogue of Timocharis of Alexandria. arg This means… D C , θ + C 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. ′ ′ | β {\displaystyle \sin(x+y)=\sin {x}\cos y+\cos x\sin y} 2 D 90 {\displaystyle R} ) Two circles 1 (r 1) and 2 (r 2) are internally/externally tangent to a circle (R) through A, B, respetively. , DEPARTMENT OF MATHEMATICS AND COMPUTER SCIENCE, THE OPEN UNIVERSITY OF SRI LANKA(OUSL), NAWALA, NUGEGODA, SRI LANKA. Ptolemy's Theorem. R . β ¯ β from which the factor D x This was a critical step in the ancient method of calculating tables of chords.. x − 2 {\displaystyle \alpha } Ptolemaic. | Article by Qi Zhu. γ , Ptolemy was often known in later Arabic sources as "the Upper Egyptian", suggesting that he may have had origins in southern Egypt. ∘ A B D {\displaystyle \beta } R is : C + . ⁡ 3 π If a quadrilateral is inscribable in a circle, then the product of the measures of its diagonals is equal to the sum of the products of the measures of the pairs of the opposite sides: A C ⋅ B D = A B ⋅ C D + A D ⋅ B C. AC\cdot BD = AB\cdot CD + AD\cdot … A B respectively. ⋅ 4 ) The theorem is named after the Greek astronomer and mathematician Ptolemy (Claudius Ptolemaeus). + Regular Pentagon inscribed in a circle, sum of distances, Ptolemy's theorem. = {\displaystyle CD=2R\sin \gamma } and θ z 2 You get the following system of equations: JavaScript is not enabled. A hexagon is inscribed in a circle. A yields Ptolemy's equality. ′ {\displaystyle \theta _{1}=90^{\circ }} 2 cos , D with , is defined by Solution: Set 's length as . ′ θ C Let ABCD be arranged clockwise around a circle in 90 = What is the value of ? ′ C 4 AC x BD = AB x CD + AD x BC Category Q.E.D. {\displaystyle \theta _{1},\theta _{2},\theta _{3}} ⋅ = C https://artofproblemsolving.com/wiki/index.php?title=Ptolemy%27s_Theorem&oldid=87049. {\displaystyle \gamma } θ ) y ( We present a proof of the generalized Ptolemys theorem, also known as Caseys theorem and its applications in the resolution of dicult geometry problems. θ {\displaystyle D'} {\displaystyle \theta _{2}=\theta _{4}} cos , β ¯ S sin ′ 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. θ B Journal of Mathematical Sciences & Mathematics Education Vol. The millenium seemed to spur a lot of people to compile "Top 100" or "Best 100" lists of many things, including movies (by the American Film Institute) and books (by the Modern Library). + 2 2 θ Ptolemy of Alexandria (~100-168) gave the name to the Ptolemy's Planetary theory which he described in his treatise Almagest. D and ) θ 2 {\displaystyle BC} Ptolemy's Theoremgives a relationship between the side lengths and the diagonals of a cyclic quadrilateral; it is the equality caseof Ptolemy's Inequality. φ 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. arg . ⋅ , Also, the sum of the products of its opposite sides is equal to the product of its diagonals. are the same D Now by using the sum formulae, y the sum of the products of its opposite sides is equal to the product of its diagonals. + θ C A z ′ 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.. Proposed Problem 291. ⁡ B [ B , it follows, Therefore, set This corollary is the core of the Fifth Theorem as chronicled by Copernicus following Ptolemy in Almagest. 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. ′ 2 , then we have 3 C = 2 θ C , it is trivial to show that both sides of the above equation are equal to. , C 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 . x {\displaystyle BD=2R\sin(\beta +\gamma )} 1 D 3 A Using Ptolemy's Theorem, . The ratio is. A {\displaystyle A'C'} θ D {\displaystyle A\mapsto z_{A},\ldots ,D\mapsto z_{D}} {\displaystyle \Gamma } B ⁡ ′ 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. | D Find the diameter of the circle. ′ C {\displaystyle BC=2R\sin \beta } and units where: It will be easier in this case to revert to the standard statement of Ptolemy's theorem: Let .J. arg Ptolemy used the theorem as an aid to creating his table of chords, a trigonometric table that he applied to astronomy. C A Then Similarly the diagonals are equal to the sine of the sum of whichever pair of angles they subtend. ⁡ D ⁡ α ( ( C = D D Then, … Ptolemy's Theorem frequently shows up as an intermediate step in problems involving inscribed figures. ⁡ {\displaystyle \theta _{1}+(\theta _{2}+\theta _{4})=90^{\circ }} La… Ptolemy's inequality is an extension of this fact, and it is a more general form of Ptolemy's theorem. ( B D y 1 This Ptolemy's Theorem Lesson Plan is suitable for 9th - 12th Grade. z The theorem that we will discuss now will be the well-known Ptolemy's theorem. But in this case, AK−CK=±AC, giving the expected result. We may then write Ptolemy's Theorem in the following trigonometric form: Applying certain conditions to the subtended angles α z {\displaystyle ABCD'} In this video we take a look at a proof Ptolemy's Theorem and how it is used with cyclic quadrilaterals. + 180 A ∘ {\displaystyle \gamma } Proof: It is known that the area of a triangle and Γ . z = x The online proof of Ptolemy's Theorem is made easier here. 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. Tangents to a circle, Secants, Square, Ptolemy's theorem. Ptolemy's Theorem gives a relationship between the side lengths and the diagonals of a cyclic quadrilateral; it is the equality case of Ptolemy's Inequality. = Solution: Consider half of the circle, with the quadrilateral , being the diameter. 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