- 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. If the cyclic quadrilateral is ABCD, then Ptolemy’s theorem is the equation AC BD = AB CD + AD BC.www2.clarku.edu/faculty/djoyce/trig/ptolemy.html
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前往 Wikipedia 查看全部内容Ptolemy's theorem - Wikipedia
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 theorem is named after the Greek astronomer and mathematician Ptolemy (Claudius Ptolemaeus). Ptolemy used the theorem as an aid to creating … 展开
Equilateral triangle
Ptolemy's Theorem yields as a corollary a pretty theorem regarding an equilateral triangle inscribed in a circle. 展开In the case of a circle of unit diameter the sides $${\displaystyle S_{1},S_{2},S_{3},S_{4}}$$ of any cyclic quadrilateral ABCD are numerically equal to the sines of the angles $${\displaystyle \theta _{1},\theta _{2},\theta _{3}}$$ and 展开
Visual proof
The animation here shows a visual demonstration of Ptolemy's theorem, based on Derrick & … 展开The equation in Ptolemy's theorem is never true with non-cyclic quadrilaterals. Ptolemy's inequality is an extension of this fact, and it is a more general form of Ptolemy's theorem. It states that, given a quadrilateral ABCD, then 展开
• Proof of Ptolemy's Theorem for Cyclic Quadrilateral
• MathPages – On Ptolemy's Theorem
• Elert, Glenn (1994). "Ptolemy's Table of Chords". E-World.
• Ptolemy's Theorem at cut-the-knot 展开CC-BY-SA 许可证中的维基百科文本 Ptolemy's Theorem - Art of Problem Solving
- Given cyclic quadrilateral extend to such that Since quadrilateral is cyclic, However, is also supplementary to so . Hence, by AA similarity and Now, note that (subtend the same arc) and so This yields However, Substituting in our expressions for and Multiplying by yields .
Ptolemy's Theorem | Brilliant Math & Science Wiki
Ptolemy's theorem states the relationship between the diagonals and the sides of a cyclic quadrilateral. It is a powerful tool to apply to problems about inscribed quadrilaterals. Let's prove this theorem.
Ptolemy's Theorem -- from Wolfram MathWorld
3 天之前 · Ptolemy's Theorem. For a cyclic quadrilateral, the sum of the products of the two pairs of opposite sides equals the product of the diagonals. (1) (Kimberling 1998, p. 223). This …
Ptolemy’s sum and difference formulas - Clark University
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. If the cyclic …
Ptolemy's Theorem - goessner
With given side and diagonal lengths, Ptolemy's theorem of a cyclic quadrilateral states: pq = ac+bd \\,. pq = ac + bd. (1) For a cyclic quadrilateral the product of the diagonal lengths is equal to the sum of the product of the length …
Theorem 1.1 (Ptolemy’s Theorem). Let A, B, C, Dbe 4 points lying in order on a circle. Then (1) ABCD+ ADBC= ACBD The same conclusion holds if the 4 points lie in order on a straight line. …
What Is Ptolemy’s Theorem? - Science ABC
2024年6月2日 · 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’. The theorem can …
Derivation / Proof of Ptolemy's Theorem for Cyclic …
Ptolemy's theorem for cyclic quadrilateral states that the product of the diagonals is equal to the sum of the products of opposite sides. From the figure below, Ptolemy's theorem can be written as $d_1 d_2 = ac + bd$
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Ptolemy’s Theorem
Ptolemy’s Theorem. Let a, b, c, and d be the lengths of consecutive sides of a cyclic quadrilateral and let x and y be the lengths of. b. y. a x c.
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