geometry Geometry (; ) is, with arithmetic, one of the oldest branches of mathematics. It is concerned with properties of space such as the distance, shape, size, and relative position of figures. A mathematician who works in the field of geometry is c ...

, Brahmagupta's theorem states that if a

cyclic quadrilateral In Euclidean geometry, a cyclic quadrilateral or inscribed quadrilateral is a quadrilateral whose vertices all lie on a single circle. This circle is called the ''circumcircle'' or ''circumscribed circle'', and the vertices are said to be ''c ...

is orthodiagonal (that is, has

perpendicular In elementary geometry, two geometric objects are perpendicular if they intersect at a right angle (90 degrees or π/2 radians). The condition of perpendicularity may be represented graphically using the '' perpendicular symbol'', ⟂. It c ...

diagonals In geometry, a diagonal is a line segment joining two vertices of a polygon or polyhedron, when those vertices are not on the same edge. Informally, any sloping line is called diagonal. The word ''diagonal'' derives from the ancient Greek δ ...

), 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 mathematician and astronomer. He is the author of two early works on mathematics and astronomy: the '' Brāhmasphuṭasiddhānta'' (BSS, "correctly established doctrine of Brahma", dated 628), a theoretical tr ...

(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 ''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 in ...

s 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°), 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 sides of equal length. Sometimes it is specified as having ''exactly'' two sides of equal length, and sometimes as having ''at least'' two sides of equal length, the latter versio ...

, 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-born Israeli-American mathematician. He was Professor Emeritus of Mathematics at the University of Iowa, and formerly research fellow at the Moscow Institute of Electronics and Math ...

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

Proof

See also

References

External links