# ptolemy's theorem trigonometry

In a quadrilateral, if the product of its diagonals is equal to the sum of the products of the pairs of the opposite sides, then the quadrilateral is inscribable. AC &= \frac{1}{AC'}\\ School Oakland University; Course Title MTH 414; Uploaded By Myxaozon911. \max \lceil BD \rceil ? But AD=BC,AB=DC,AC=DBAD= BC, AB = DC, AC = DBAD=BC,AB=DC,AC=DB since ABDCABDCABDC is a rectangle. Ptolemy: Dost thou see that all the red lines have the lengths in whole integers? 85.60 A trigonometric proof of Ptolemy’s theorem - Volume 85 Issue 504 - Ho-Joo Lee Skip to main content We use cookies to distinguish you from other users and to provide you with a better experience on our websites. In order to prove his sum and difference forumlas, Ptolemy first proved what we now call Ptolemy’s theorem. The latter serves as a foundation of Trigonometry, the branch of mathematics that deals with relationships between the sides and angles of a triangle. \end{aligned}AB⋅CD+AD⋅BC​=CE⋅DB+AE⋅DB=(CE+AE)DB=CA⋅DB.​. Ptolemy: Now if the equilateral triangle has a side length of 13, what is the sum of the three red lengths combined? 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. Ptolemy’s Theorem states, ‘For a quadrilateral inscribed in a circle, the sum of the products of each pair of opposite sides is equal to the product of its two diagonals’. AB &= \frac{1}{AB'}\\ \hspace {1.5cm} Instead, we’ll use Ptolemy’s theorem to derive the sum and difference formulas. Then since ∠ABE=∠CBK\angle ABE= \angle CBK∠ABE=∠CBK and ∠CAB=∠CDB,\angle CAB= \angle CDB,∠CAB=∠CDB, △ABE≈△BDC⟺ABDB=AECD⟺CD⋅AB=DB⋅AE. File:Ptolemy Rectangle.svg … 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: AC⋅BD=AB⋅CD+AD⋅BC.AC\cdot BD = AB\cdot CD + AD\cdot BC.AC⋅BD=AB⋅CD+AD⋅BC. If the vertices in clockwise order are A, B, C and D, this means that the triangles ABC, BCD, CDA and DAB all have the same circumcircle and hence the same circumradius. Ptolemy’s Theorem is a powerful geometric tool. Ptolemy's Incredible Theorem - Part 1 Ptolemy was an ancient astronomer, geographer, and mathematician who lived from (c. AD 100 – c. 170). C'D' + B'C' &\geq B'D', What is SOHCAHTOA . □BC^2 = AB^2 + AC^2. Proofs of Ptolemy’s Theorem can be found in Aaboe, 1964, Berggren, 1986, and Katz, 1998. Sign up to read all wikis and quizzes in math, science, and engineering topics. His contributions to trigonometry are especially important. CD &= \frac{C'D'}{AC' \cdot AD'}\\ The table of chords, created by the Greek astronomer, geometer, and geographer Ptolemy in Egypt during the 2nd century AD, is a trigonometric table in Book I, chapter 11 of Ptolemy's Almagest, a treatise on mathematical astronomy. In the case of a circle of unit diameter the sides of any cyclic quadrilateral ABCD are numerically equal to the sines of the angles and which they subtend. This preview shows page 5 - 7 out of 7 pages. Ptolemy's theorem states the relationship between the diagonals and the sides of a cyclic quadrilateral. Pupil: Indeed, master! Therefore sin ∠ACB cos α. Triangle ABDABDABD is similar to triangle IBCIBCIBC, so ABIB=BDBC=ADIC  ⟹  AD⋅BC=BD⋅IC\frac{AB}{IB}=\frac{BD}{BC}=\frac{AD}{IC} \implies AD \cdot BC = BD \cdot ICIBAB​=BCBD​=ICAD​⟹AD⋅BC=BD⋅IC and ABBD=IBBC\frac{AB}{BD}=\frac{IB}{BC}BDAB​=BCIB​. □_\square□​. Therefore, Ptolemy's inequality is true. Likewise, AD = 2 cos β. & = CA\cdot DB. Ptolemy's Theorem frequently shows up as an intermediate step … It was the earliest trigonometric table extensive enough for many practical purposes, … Few details of Ptolemy's life are known. The equality occurs when III lies on ACACAC, which means ABCDABCDABCD is inscribable. (2), Therefore, from (1)(1)(1) and (2),(2),(2), we have, AB⋅CD+AD⋅BC=CE⋅DB+AE⋅DB=(CE+AE)DB=CA⋅DB.\begin{aligned} World's Best PowerPoint Templates - CrystalGraphics offers more PowerPoint templates than anyone else in the world, with over 4 million to choose from. Determine the length of the line segment formed when PQ‾\displaystyle \overline{PQ}PQ​ is extended from both sides until it reaches the circle. It is essentially equivalent to a table of values of the sine function. Sign up, Existing user? For example, take AD to be a diameter, α to be ∠BAD, and β to be ∠CAD, then you can directly show the difference formula for sines. Ptolemy was often known in later Arabic sources as "the Upper Egyptian", suggesting that he may have had origins i… In wh… Sine, Cosine, and Ptolemy's Theorem. We won't prove Ptolemy’s theorem here. We’ll follow Ptolemy’s proof, but modify it slightly to work with modern sines. Similarly the diagonals are equal to the sine of the sum of whichever pairof angles they subtend. A B D C Figure 1: Cyclic quadrilateral ABCD Proof. □_\square□​. ryT proving it by yourself rst, then come back. A cyclic quadrilateral ABCDABCDABCD is constructed within a circle such that AB=3,BC=6,AB = 3, BC = 6,AB=3,BC=6, and △ACD\triangle ACD△ACD is equilateral, as shown to the right. Ptolemy's Theorem. Euclid’s proposition III.20 says that the angle at the center of a circle twice the angle at the circumference, therefore ∠BOC equals 2α. & = (CE+AE)DB \\ 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 BC. Ptolemy's theorem implies the theorem of Pythagoras. In trigonometry, the law of cosines (also known as the cosine formula, cosine rule, or al-Kashi's theorem) relates the lengths of the sides of a triangle to the cosine of one of its angles.Using notation as in Fig. \ _\squareBC2=AB2+AC2. Let α be ∠BAC. Triangle ABC is a right triangle by Thale’s theorem (Euclid’s proposition III.31: an angle in a semicircle is right). In Euclidean geometry, Ptolemys theorem is a relation between the four sides and two diagonals of a cyclic quadrilateral. ⁡. Then, he created a mathematical model for each planet. He also applied fundamental theorems in spherical trigonometry (apparently discovered half a century earlier by Menelaus of Alexandria) to the solution of many basic astronomical problems. Though many problems may initially appear impenetrable to the novice, most can be solved using only elementary high school mathematics techniques. As you know, three points determine a circle, so the fourth vertex of the quadrilateral is constrained, … subsidy of trigonometry or vector algebra just a little bit. \end{aligned}AB⋅CD+AD⋅BCAB′1​⋅AC′⋅AD′C′D′​+AD′1​⋅AB′⋅AC′B′C′​C′D′+B′C′​≥BD⋅AC≥AC′1​⋅AB′⋅AD′B′D′​≥B′D′,​, which is true by triangle inequality. AC BD= AB CD+ AD BC. Let O to be the center of a circle of radius 1, and take one of the lines, AC, to be a diameter of the circle. Already have an account? We already know AC = 2. Ptolemy used the theorem as an aid to creating his table of chords, a trigonometric table that he applied to astronomy. • Menelaus’s theorem: this result is dual to Ceva’s theorem (and its converse) in the sense that it gives a way to check when three points are on a line (collinearity) in Log in here. The theorem is named after the Greek astronomer and mathematician Ptolemy. \qquad (1)△EBC≈△ABD⟺DBCB​=ADCE​⟺AD⋅CB=DB⋅CE.(1). Ptolemy's Theorem and Familiar Trigonometric Identities. Key features: * Gradual progression in problem difficulty … Ptolemy's theorem - Wikipedia wikimedia.org. It is a powerful tool to apply to problems about inscribed quadrilaterals. In Trigonometric Delights (Chapter 6), Eli Maor discusses this delightful theorem that is so useful in trigonometry. Pages 7. https://brilliant.org/wiki/ptolemys-theorem/. Thus, the sine of α is half the chord of ∠BOC, so it equals BC/2, and so BC = 2 sin α. Bidwell, James K. School Science and Mathematics, v93 n8 p435-39 Dec 1993. We can prove the Pythagorean theorem using Ptolemy's theorem: Prove that in any right-angled triangle △ABC\triangle ABC△ABC where ∠A=90∘,\angle A = 90^\circ,∠A=90∘, AB2+AC2=BC2.AB^2 + AC^2 = BC^2.AB2+AC2=BC2. ( β + γ) sin. The theorem was mentioned in Chapter 10 of Book 1 of Ptolemy’s Almagest and relates the four sides of a cyclic quadrilateral (a quadrilateral with all four vertices on a single circle) to its diagonals. ⓘ Ptolemys theorem. The proposition will be proved if AC⋅BD=AB⋅CD+AD⋅BC.AC\cdot BD = AB\cdot CD + AD\cdot BC.AC⋅BD=AB⋅CD+AD⋅BC. Stack Exchange network consists of 176 Q&A communities including Stack Overflow, the largest, most trusted online community for developers to learn, share their knowledge, and build their careers.. Visit Stack Exchange In this video we take a look at a proof Ptolemy's Theorem and how it is used with cyclic quadrilaterals. top; sohcahtoa; Unit Circle; Trig Graphs; Law of (co)sines; Miscellaneous; Trig Graph Applet. Proofs of ptolemys theorem can be found in aaboe 1964. Let EEE be a point on ACACAC such that ∠EBC=∠ABD=∠ACD, \angle EBC = \angle ABD = \angle ACD,∠EBC=∠ABD=∠ACD, then since ∠EBC=∠ABD \angle EBC = \angle ABD ∠EBC=∠ABD and ∠BCA=∠BDA,\angle BCA= \angle BDA,∠BCA=∠BDA, △EBC≈△ABD⟺CBDB=CEAD⟺AD⋅CB=DB⋅CE. Recall that the sine of an angle is half the chord of twice the angle. This was the precursor to the modern sine function. Hence, AB = 2 cos α. Claudius Ptolemy was the first to use trigonometry to calculate the positions of the Sun, the Moon, and the planets. Ptolemy’s theorem: For a cyclic quadrilateral (that is, a quadrilateral inscribed in a circle), the product of the diagonals equals the sum of the products of the opposite sides. App; Gifs ; applet on its own page SOHCAHTOA . File:Ptolemy Theorem az.svg - Wikimedia Commons wikimedia.org. Integrates the sum, difference, and multiple angle identities into an examination of Ptolemy's Theorem, which states that the sum of the products of the lengths of the opposite sides of a quadrilateral inscribed in a circle is equal to the product … We’ll interpret each of the lines AC, BD, AB, CD, AD, and BC in terms of sines and cosines of angles. New user? Originally, the Theorem of Menelaos applied to complete spherical quadrilaterals served this purpose virtually single-handedly, but it would be followed by results derived later, such as the Rule of Four Quantities and the Spherical Law of … ( α + γ) This statement is equivalent to the part of Ptolemy's theorem that says if a quadrilateral is inscribed in a circle, then the product of the diagonals equals the sum of the products of the opposite sides. He lived in Egypt, wrote in Ancient Greek, and is known to have utilised Babylonian astronomical data. 1, the law of cosines states = + − ⁡, where γ denotes the angle contained between sides of lengths a and b and opposite the side of length c. Ptolemys Theorem - YouTube ytimg.com. We still have to interpret AB and AD. You can use these identities without knowing why they’re true. I will now present these corollaries and the subsequent proofs given by Ptolemy. If you replace β by −β, you’ll get the difference formula. PPP and QQQ are points on AB‾\overline{AB}AB and CD‾ \overline{CD}CD, respectively, such that AP‾=6\displaystyle \overline{AP}=6AP=6, DQ‾=7\displaystyle \overline{DQ}=7DQ​=7, and PQ‾=27.\displaystyle \overline{PQ}=27.PQ​=27. Thus proven. In the language of Trigonometry, Pythagorean Theorem reads $\sin^{2}(A) + \cos^{2}(A) = 1,$ The right and left-hand sides of the equation reduces algebraically to form the same kind of expression. If the cyclic quadrilateral is ABCD, then Ptolemy’s theorem is the equation. Proof of Ptolemy’s Theorem | Advanced Math Class at ... wordpress.com. Such an extraordinary point! AC ⋅BD = AB ⋅C D+AD⋅ BC. Ptolemy's theorem - Wikipedia wikimedia.org. Ptolemy's Theoremgives a relationship between the side lengths and the diagonals of a cyclic quadrilateral; it is the equality caseof Ptolemy's Inequality. Sine, Cosine, and Ptolemy's Theorem; arctan(1) + arctan(2) + arctan(3) = π; Trigonometry by Watching; arctan(1/2) + arctan(1/3) = arctan(1) Morley's Miracle; Napoleon's Theorem; A Trigonometric Solution to a Difficult Sangaku Problem; Trigonometric Form of Complex Numbers; Derivatives of Sine and Cosine; ΔABC is right iff sin²A + sin²B + sin²C = 2 δ = sin. Ptolemy's Theorem | Brilliant Math & Science Wiki cloudfront.net. 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. Winner of the Standing Ovation Award for “Best PowerPoint Templates” from Presentations Magazine. For instance, Ptolemy’s table of the lengths of chords in a circle is the earliest surviving table of a trigonometric function. Let B′,C′,B', C',B′,C′, and D′D'D′ be the resultant of inverting points B,C,B, C,B,C, and DDD about this circle, respectively. \frac{1}{AB'} \cdot \frac{C'D'}{AC' \cdot AD'} + \frac{1}{AD'} \cdot \frac{B'C'}{AB' \cdot AC'} &\geq \frac{1}{AC'} \cdot \frac{B'D'}{AB' \cdot AD'}\\\\ \end{aligned}ABCDADBCACBD​=AB′1​=AC′⋅AD′C′D′​=AD′1​=AB′⋅AC′B′C′​=AC′1​=AB′⋅AD′B′D′​.​, AB⋅CD+AD⋅BC≥BD⋅AC1AB′⋅C′D′AC′⋅AD′+1AD′⋅B′C′AB′⋅AC′≥1AC′⋅B′D′AB′⋅AD′C′D′+B′C′≥B′D′,\begin{aligned} In a cyclic quadrilateral the product of the diagonals is equal to the sum of the products of the pairs of opposite sides. 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'. In spherical astronomy, the Ptolemaic strategy is to operate mainly on the surface of the sphere by using theorems of spherical trigonometry per se. (1)\triangle EBC \approx \triangle ABD \Longleftrightarrow \dfrac{CB}{DB} = \dfrac{CE}{AD} \Longleftrightarrow AD\cdot CB = DB\cdot CE. \qquad (2)△ABE≈△BDC⟺DBAB​=CDAE​⟺CD⋅AB=DB⋅AE. The proof depends on properties of similar triangles and on the Pythagorean theorem. AB \cdot CD + AD \cdot BC &\geq BD \cdot AC\\ Another proof requires a basic understanding of properties of inversions, especially those relevant to distance. We won't prove Ptolemy’s theorem here. □​. max⌈BD⌉? You could investigate how Ptolemy used this result along with a few basic triangles to compute his entire table of chords. Therefore, BC2=AB2+AC2. If EEE is the intersection point of both diagonals of ABCDABCDABCD, what is the length of ED,ED,ED, the blue line segment in the diagram? If you replace certain angles by their complements, then you can derive the sum and difference formulas for cosines. With this theorem, Ptolemy produced three corollaries from which more chord lengths could be calculated: the chord of the difference of two arcs, the chord of half of an arc, and the chord of the sum of two arcs. We’ll derive this theorem now. Alternatively, you can show the other three formulas starting with the sum formula for sines that we’ve already proved. SOHCAHTOA HOME. He is most famous for proposing the model of the "Ptolemaic system", where the Earth was considered the center of the universe, and the stars revolve around it. Forgot password? The line segment AB is twice the sine of ∠ACB. BC &= \frac{B'C'}{AB' \cdot AC'}\\ Ptolemy's Theorem. Let III be a point inside quadrilateral ABCDABCDABCD such that ∠ABD=∠IBC\angle ABD = \angle IBC∠ABD=∠IBC and ∠ADB=∠ICB\angle ADB = \angle ICB∠ADB=∠ICB. 2 Ptolemy's Theorem - The key of this Handout Ptolemy's Theorem If ABCD is a (possibly degenerate) cyclic quadrilateral, then jABjjCDj+jADjjBCj= jACjjBDj. They'll give your presentations a professional, memorable appearance - the kind of sophisticated look … If you’re interested in why, then keep reading, otherwise, skip on to the next page. The incentres of these four triangles always lie on the four vertices of a rectangle; these four points plus the twelve excentres form a rectangular 4x4 grid. Using the distance properties of inversion, we have, AB=1AB′CD=C′D′AC′⋅AD′AD=1AD′BC=B′C′AB′⋅AC′AC=1AC′BD=B′D′AB′⋅AD′.\begin{aligned} Let ABDCABDCABDC be a random rectangle inscribed in a circle. That’s half of ∠COD, so ∠BAC=∠BDC. Then α + β is ∠BAD, so BD = 2 sin (α + β). Consider a circle of radius 1 centred at AAA. We may then write Ptolemy's Theorem in the following trigonometric form: Applying certain conditions to the subtended angles and it is possible to derive a number of important corollaries using the above as our starting point. AD⋅BC=AB⋅DC+AC⋅DB.AD\cdot BC = AB\cdot DC + AC\cdot DB.AD⋅BC=AB⋅DC+AC⋅DB. sin β equals CD/2, and CD = 2 sin β. I will also derive a formula from each corollary that can be used to calc… After dividing by 4, we get the addition formula for sines. Spoilers ahead! Ptolemy used it to create his table of chords. The theorem refers to a quadrilateral inscribed in a circle. \hspace{1.5cm}. ABCDABCDABCD is a cyclic quadrilateral with AB‾=11\displaystyle \overline{AB}=11AB=11 and CD‾=19\displaystyle \overline{CD}=19CD=19. 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