Brahmagupta theorem

Brahmagupta theorem

Brahmagupta's theorem is a result in geometry. It states that if a cyclic quadrilateral has perpendicular diagonals, then the perpendicular to a side from the point of intersection of the diagonals always bisects the opposite side. It is named after the Indian mathematician Brahmagupta.

More specifically, let "A", "B", "C" and "D" be four points on a circle such that the lines "AC" and "BD" are perpendicular. Denote the intersection of "AC" and "BD" by "M". Drop the perpendicular from "M" to the line "BC", calling the intersection "E". Let "F" be the intersection of the line "EM" and the edge "AD". Then, the theorem states that "F" is in the middle of "AD".


We need to prove that "AF" = "FD". We will prove that both "AF" and "FD" are in fact equal to "FM".

To prove that "AF" = "FM", first note that the angles "FAM" and "CBM" are equal, because they are inscribed angles that intercept the same arc of the circle. Furthermore, the angles "CBM" and "CME" are both complementary to angle "BCM" (i.e., they add up to 90°). Finally, the angles "CME" and "FMA" are the same. Hence, "AFM" is an isosceles triangle, and thus the sides "AF" and "FM" are equal.

The proof that "FD" = "FM" goes similarly. The angles "FDM", "BCM", "BME" and "DMF" are all equal, so "DFM" is an isosceles triangle, so "FD" = "FM". It follows that "AF" = "FD", as the theorem claims.

External links

*MathWorld|urlname=BrahmaguptasTheorem|title=Brahmagupta's theorem
* [ Brahmagupta's Theorem] at cut-the-knot

Wikimedia Foundation. 2010.

Игры ⚽ Поможем решить контрольную работу

Look at other dictionaries:

  • Brahmagupta — (audio|Brahmagupta pronounced.ogg|listen) (598–668) was an Indian mathematician and astronomer. Life and work Brahmagupta was born in 598 CE in Bhinmal city in the state of Rajasthan of northwest India. He likely lived most of his life in… …   Wikipedia

  • Brahmagupta's formula — In geometry, Brahmagupta s formula finds the area of any quadrilateral given the lengths of the sides and some of their angles. In its most common form, it yields the area of quadrilaterals that can be inscribed in a circle. Basic form In its… …   Wikipedia

  • Proofs of Fermat's theorem on sums of two squares — Fermat s theorem on sums of two squares asserts that an odd prime number p can be expressed as: p = x^2 + y^2with integer x and y if and only if p is congruent to 1 (mod 4). The statement was announced by Fermat in 1640, but he supplied no proof …   Wikipedia

  • Théorème de Brahmagupta — et implique AF = FD En mathématiques, le théorème de Brahmagupta donne une condition nécessaire sur la …   Wikipédia en Français

  • Eieraufgabe des Brahmagupta — Die Eieraufgabe des Brahmagupta[1], im Englischen auch als Egg Basket Problem[2] bekannt, ist eine als Anwendungsproblem eingekleidete zahlentheoretische Aufgabe. Hierbei erfüllt die Anzahl der Eier in einem Korb eine Reihe von Bedingungen,… …   Deutsch Wikipedia

  • Satz von Brahmagupta — Der Satz von Brahmagupta ist eine Aussage in der euklidischen Geometrie über Streckenverhältnisse in bestimmten Sehnenvierecken. Wenn …   Deutsch Wikipedia

  • Chinese remainder theorem — The Chinese remainder theorem is a result about congruences in number theory and its generalizations in abstract algebra. In its most basic form it concerned with determining n, given the remainders generated by division of n by several numbers.… …   Wikipedia

  • Fermat's theorem on sums of two squares — In number theory, Pierre de Fermat s theorem on sums of two squares states that an odd prime p is expressible as:p = x^2 + y^2,,with x and y integers, if and only if:p equiv 1 pmod{4}.The theorem is also known as Thue s Lemma, after Axel Thue.For …   Wikipedia

  • List of mathematics articles (B) — NOTOC B B spline B* algebra B* search algorithm B,C,K,W system BA model Ba space Babuška Lax Milgram theorem Baby Monster group Baby step giant step Babylonian mathematics Babylonian numerals Bach tensor Bach s algorithm Bachmann–Howard ordinal… …   Wikipedia

  • Timeline of mathematics — A timeline of pure and applied mathematics history. Contents 1 Before 1000 BC 2 1st millennium BC 3 1st millennium AD 4 1000–1500 …   Wikipedia

Share the article and excerpts

Direct link
Do a right-click on the link above
and select “Copy Link”