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What is Ptolemy's theorem used for?

What is the triangle inequality theorem in geometry?

triangle inequality, in Euclidean geometry, theorem that the sum of any two sides of a triangle is greater than or equal to the third side; in symbols, a + b ≥ c. In essence, the theorem states that the shortest distance between two points is a straight line.

When was Ptolemy's theorem discovered?

Every right triangle (in fact every triangle) can be circumscribed by a circle. Six centuries later, Greek mathematician Claudius Ptolemy (ca. 90–ca. 168 CE) discovered a relationship between the side lengths and the diagonals of any quadrilateral inscribed in a circle.

What is Ptolemy's geocentric model?

Ptolemy placed the Earth at the centre of his geocentric model. ... He believed that the Moon was orbiting on a sphere closest to the Earth, followed by Mercury, then Venus and then the Sun. Beyond the Sun were a further three spheres on which Mars, then Jupiter and then Saturn orbited the Earth.

Who is known as father of trigonometry?

The first known table of chords was produced by the Greek mathematician Hipparchus in about 140 BC. Although these tables have not survived, it is claimed that twelve books of tables of chords were written by Hipparchus. This makes Hipparchus the founder of trigonometry.

image-What is Ptolemy's theorem used for?
image-What is Ptolemy's theorem used for?
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How do you find a triangle inequality?

According to the triangle inequality theorem, the sum of any two sides of a triangle is greater than or equal to the third side of a triangle. This statement can symbolically be represented as; a + b > c. a + c > b.

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What is half the product of the diagonals?

ANSWER. The area of a rhombus is half the product of its diagonals.Jun 30, 2020

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What is the cyclic quadrilateral theorem?

The ratio between the diagonals and the sides can be defined and is known as Cyclic quadrilateral theorem. If there's a quadrilateral which is inscribed in a circle, then the product of the diagonals is equal to the sum of the product of its two pairs of opposite sides.

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What does Ptolemy mean?

1 : of or relating to the second century geographer and astronomer Ptolemy of Alexandria and especially to his belief that the earth is at the center of the universe with the sun, moon, and planets revolving around it the Ptolemaic system.

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How strict is Ptolemy's inequality?

  • For any other four points, Ptolemy's inequality is strict. Ptolemy's inequality holds more generally in any inner product space, and whenever it is true for a real normed vector space, that space must be an inner product space.

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Does Ptolemy's theorem apply to all four points in the plane?

  • However, the theorem applies more generally to any four points; it is not required that the quadrilateral they form be convex, simple, or even planar. For points in the plane, Ptolemy's inequality can be derived from the triangle inequality by an inversion centered at one of the four points.

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What are the Ptolemaic spaces?

  • The spaces where it remains valid are called the Ptolemaic spaces; they include the inner product spaces, Hadamard spaces, and shortest path distances on Ptolemaic graphs . Ptolemy's inequality is often stated for a special case, in which the four points are the vertices of a convex quadrilateral, given in cyclic order.

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How do you solve the triangle inequality with three products?

  • Thus the three product terms in the inequality can be additively permuted to put any one of them on the right side of the inequality, so the three products of opposite sides or of diagonals of any one of the quadrilaterals must obey the triangle inequality.

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