How did Menelaus contribute to trigonometry?
How did Menelaus contribute to trigonometry?
In Book I he established the basis for a mathematical treatment of spherical triangles analogous to Euclid’s treatment of plane triangles. Furthermore, he originated the use of arcs of great circles instead of arcs of parallel circles on the sphere, a major turning point in the development of spherical trigonometry.
What is the objective of Menelaus theorem?
Menelaus’ theorem deals with the collinearity of points on each of the three sides (extended when necessary) of a triangle. Calculation: Consider ΔABC, Then D, E and F are, respectively, points on the side CA, AB and BC, and by construction are collinear.
Is there any difference between Ceva theorem and Menelaus theorem?
Ceva’s Theorem states that if the three Cevians of a triangle are concurrent then the previous statement holds. On the other hand, Menelaus’ Theorem states that if points D, E, and F on the sides BC, CA, and AB of triangle ABC are collinear, then the previous statement holds.
How do you prove the Menelaus theorem?
A standard proof is as follows: First, the sign of the left-hand side will be negative since either all three of the ratios are negative, the case where the line DEF misses the triangle (lower diagram), or one is negative and the other two are positive, the case where DEF crosses two sides of the triangle.
Who made spherical trigonometry?
Nasīr al-Dīn al-Tūsī
In the 13th century, Nasīr al-Dīn al-Tūsī was the first to treat trigonometry as a mathematical discipline independent from astronomy, and he developed spherical trigonometry into its present form.
What does Menelaus theorem prove?
Menelaus’ theorem relates ratios obtained by a line cutting the sides of a triangle. The converse of the theorem (i.e. three points on a triangle are collinear if and only if they satisfy certain criteria) is also true and is extremely powerful in proving that three points are collinear.
How are Ceva’s theorem and the theorem of Menelaus similar?
In their most basic form, Ceva’s Theorem and Menelaus’s Theorem are simple formulas of triangle geometry. To state them, we require some definitions. Three or more line segments in the plane are concurrent if they have a common point of intersection. 1The reverse is true, but complicated.
Why are medians of a triangle concurrent?
Proof that medians are concurrent Since the median of any side of the triangle will always be contained on the segment that forms the side of the triangle, then the segment connecting that median to the opposite vertex will also necessarily be on the interior of the triangle.
What is Girard’s theorem?
Girard’s theorem states that the area of a spherical triangle is given by the spherical excess: , where the interior angles of the triangle are , , , and the radius of the sphere is 1. Rewriting the formula in terms of the exterior angles ‘, ‘, and ‘ gives the equivalent formula .
What are Midsegments of a triangle?
A midsegment of a triangle is a segment that connects the midpoints of two sides of a triangle.
What is Napier’s rule spherical triangle?
In the Napier’s circle, the sine of any middle part is equal to the product of the tangents of its adjacent parts. Spherical triangle can have one or two or three 90° interior angle. Spherical triangle is said to be right if only one of its included angle is equal to 90°.
What is Napier’s analogy?
Definition of Napier’s analogies : four formulas giving the tangent of half the sum or difference of two of the angles or sides of a spherical triangle in terms of the others.
What is Menelaus’s theorem?
Menelaus’s theorem, named for Menelaus of Alexandria, is a proposition about triangles in plane geometry. Suppose we have a triangle ABC, and a transversal line that crosses BC, AC, and AB at points D, E, and F respectively, with D, E, and F distinct from A, B, and C.
What does Menelaus say about the length of a triangle?
Menelaus: Mark this, lad. Point E E also divide the triangle’s sides into integer lengths. Pupil: O, so true, master! Any length between any two of those points is always a whole integer! Menelaus: Then thou shalt tell me.
What is Menelaus’theorem?
In Almagest, Ptolemy applies the theorem on a number of problems in spherical astronomy. During the Islamic Golden Age, Muslim scholars devoted a number of works that engaged in the study of Menelaus’s theorem, which they referred to as “the proposition on the secants” ( shakl al-qatta’ ).
Is the converse part of the theorem?
The converse is often included as part of the theorem. The theorem is very similar to Ceva’s theorem in that their equations differ only in sign.