geometry Geometry (; ) is a branch of mathematics concerned with properties of space such as the distance, shape, size, and relative position of figures. Geometry is, along with arithmetic, one of the oldest branches of mathematics. A mathematician w ...

, Brahmagupta's theorem states that if a

cyclic quadrilateral In geometry, a cyclic quadrilateral or inscribed quadrilateral is a quadrilateral (four-sided polygon) whose vertex (geometry), vertices all lie on a single circle, making the sides Chord (geometry), chords of the circle. This circle is called ...

is orthodiagonal (that is, has

perpendicular In geometry, two geometric objects are perpendicular if they intersect at right angles, i.e. at an angle of 90 degrees or π/2 radians. The condition of perpendicularity may be represented graphically using the '' perpendicular symbol'', � ...

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 Indian mathematics emerged in the Indian subcontinent from 1200 BCE until the end of the 18th century. In the classical period of Indian mathematics (400 CE to 1200 CE), important contributions were made by scholars like Aryabhata, Brahmagupta, ...

Brahmagupta Brahmagupta ( – ) was an Indian Indian mathematics, mathematician and Indian astronomy, astronomer. He is the author of two early works on mathematics and astronomy: the ''Brāhmasphuṭasiddhānta'' (BSS, "correctly established Siddhanta, do ...

(598-668). Coxeter, H. S. M.; Greitzer, S. L.: ''Geometry Revisited''. Washington, DC: Math. Assoc. Amer., p. 59, 1967 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 the

midpoint In geometry, the midpoint is the middle point of a line segment. It is equidistant from both endpoints, and it is the centroid both of the segment and of the endpoints. It bisects the segment. Formula The midpoint of a segment in ''n''-dim ...

''AD''.

Proof

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 angle In geometry, an inscribed angle is the angle formed in the interior of a circle when two chords intersect on the circle. It can also be defined as the angle subtended at a point on the circle by two given points on the circle. Equivalently, an ...

s that intercept the same arc of the circle (CD). Furthermore, the angles ''CBM'' and ''CME'' are both complementary to angle ''BCM'' (i.e., they add up to 90°), and are therefore equal. Finally, the angles ''CME'' and ''FMA'' are the same. Hence, ''AFM'' is an

isosceles triangle In geometry, an isosceles triangle () is a triangle that has two Edge (geometry), sides of equal length and two angles of equal measure. Sometimes it is specified as having ''exactly'' two sides of equal length, and sometimes as having ''at le ...

, 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.

References

External links

Brahmagupta's Theorem
at

cut-the-knot Alexander Bogomolny (January 4, 1948 July 7, 2018) was a Soviet Union, Soviet-born Israeli Americans, Israeli-American mathematician. He was Professor Emeritus of Mathematics at the University of Iowa, and formerly research fellow at the Moscow ...

*{{MathWorld, urlname=BrahmaguptasTheorem, title=Brahmagupta's theorem Brahmagupta Theorems about quadrilaterals and circles Articles containing proofs

Proof

See also

References

External links