india's narayan-pandit[1]
TRANSCRIPT
05/03/23 1India's Contribution to Geometry
India's Contribution to Geometry
Narayan PanditPresented by:- Mrs . Geeta Ghormade Innovation & Research Cell , MGS Nagpur
05/03/23 2India's Contribution to Geometry
Scripts: 1) An arithmetical treatise - Ganit Kaumudi, 2) An algebraic treatise – Bijaganita Vatamsa
•Lived in 14 th century AD in the period of (1340 - 1400)•Mathematician of medieval period .•Kerala School of Mathematics
Narayan Pandit
Ganit Kaumudi
05/03/23 India's Contribution to Geometry 3
Chapter 4 - Triangles, quadrilaterals, circle, their areas, formation of integral triangle and quadrilaterals, cyclic quadrilaterals
05/03/23 4India's Contribution to Geometry
Formulae for Triangle
• Area of triangle =• If a, b, c are sides of the triangle and s is
semi perimeter i.e. 2 s = a + b + c then Area of triangle = [s (s-a) (s-b) (s-c)]1/2
• Circum radius =
• Radius of inscribed circle =
2HeightBase
altitudesidesofproduct
2
PerimetrerArea2
05/03/23 5India's Contribution to Geometry
Narayana’s Results for Circum radius
1) R = [ BC2+ {(AD2 - BD × DC)/AD}2 ]1/2
2) R =
A
B CD
21
21
altitudesofproductflanksofproductdiagonalsofoduct Pr
05/03/23 India's Contribution to Geometry 6
Narayana’s Results for Circumradius
R = 21
altitudesofproductflanksofproductdiagonalsofoduct Pr
From ADB
R = =
altitudesidesofoduct
2Pr
12.PBDAD
From ACB
R = =
altitudesidesofoduct
2Pr
22.PBCAC
21....
21
ppBCADBDACR
05/03/23 India's Contribution to Geometry 7
Area of Triangle
The area of triangle is the product of sides divided by 4 times the circum radius
RcbaA
4
BC = a CA = bAB = c
‘O’ is the centre of circum - circle R = Circum radius
A
B C
O
05/03/23 India's Contribution to Geometry 8
A
B C
E
D
ORcbaAPROOF
4:
)1(2
.21
.21
aAAD
ADaA
ADBCABCofArea
05/03/23 India's Contribution to Geometry 9
similarareACEandADB
CD
arcsametheininscribedAnglesEBACEandADBIn
2
A
B C
E
D
O
05/03/23 India's Contribution to Geometry 10
A
B C
E
D
O
RabcA
FromaA
bRc
bAD
Rc
csstACAD
AEAB
similarareACEandADB
4
)1(2.12
2
)(
05/03/23 India's Contribution to Geometry 11
If the altitude is produced to meet the circum-circle , the portion beneath the base can be calculated using the sutra
Meaning:- The lower part of the altitude which touches the circum-circle is product of the parts of the base divided by the altitude
DE = (BD × DC) / AD.BE = (BD × AC ) / AD.CE = (CD × AB) / AD
A
B C
E
DO
Third Diagonal of a quadrilateral
12
Definition:- When the top side and the flank side of a quadrilateral are interchanged a third diagonal is generated called as a ‘para’
In a quadrilateral ABCD interchange the sides CD & CB.Select a point P on the circum-circle such that
BP = CD and DP = BC
Then AP is third diagonal
05/03/2313
India's Contribution to Geometry
Area of a Cyclic Quadrilateral
The area of cyclic quadrilateral is given by the product of three diagonals divided by twice the circum -diameter
In quad .ABCD AC and BD are original diagonals .AP is third diagonal.
DAPBDACABCDA
2)(
05/03/23 India's Contribution to Geometry 14
)..(4
)..(4
4..
4..
)()(
ABDPBPADRAC
ABBCCDADRAC
RABCBAC
RADCDAC
ACBAACDAralquadrilateofArea
Area of a Cyclic Quadrilateral
05/03/23 India's Contribution to Geometry 15
RBDAPAC
BDAPRAC
ABDPBPADRAC
4..
).(4
)..(4
Ptolemy’s TheoremIn a cyclic quadrilateral , sum of product of opposite sides = product of diagonalsAD . BP + DP . AB =AP . BD
05/03/23 India's Contribution to Geometry 16
Area of a cyclic Quadrilateral
When the diagonal is multiplied by the sum of products of the sides about the other diagonal and divided by four times the circum –radius , that will be the area of isosceles trapezia and other cyclic quadrilateral
05/03/23 India's Contribution to Geometry17
When the diagonal is multiplied by the sum of products of the sides about the other diagonal and divided by four times the circum –radius , that will be the area of isosceles trapezia and other cyclic quadrilateral
A B
CD
L
K
RDCBCABADBD
RABBCDCADAC
ABCDA
4)..(
4)..()(
Area of a cyclic Quadrilateral
05/03/23 India's Contribution to Geometry 18
Area of a cyclic Quadrilateral
A B
CD
L
K
RBCABDCADAC
RBCABAC
RDCADAC
BKACDLACABCAADCAABCDA
4)..(2
..22
..2
2.
2.
)()()(
05/03/23 India's Contribution to Geometry 19
Diagonals of Cyclic Quadrilateral
• AB = a, BC = b, CD = c, DA = d, DB = x, AC = y, AP = z (Third Diagonal)
• Diagonal AC = [(ac+bd)(ad+bc)/(ab+cd)]1/2
• Diagonal BD = [(ad+bc)(ac+bd)/(ab+cd)]1/2
• Diagonal AP = [(ab+cd)(ad+bc)/ ac+bd)]1/2
05/03/23 India's Contribution to Geometry 20
Formation of Integral Triangle(Rational Triangle)
Rational Triangle :- A triangle in the Euclidean plane such that all three sides measured relative to each other are integer .
He gave a rule of finding rational triangles whose sides differ by one unit of length
( 3 , 4 , 5 ) ( 13 ,14 ,15 ) (51 , 52 , 53 ) (193 , 194 , 195 ) (2701 ,2702 , 2703 )
05/03/23 India's Contribution to Geometry 21
Ganita Kaumudi
Ganit Kaumudi is for removing the darkness and increasing the knowledge of mathematics which is like a sea and which is the life of many people.
Ganita Kaumudi
May this Ganita Kaumudi with its pleasant light puff up our pride in Bharatiya Ganita and empower us to touch new horizons in the subject.
References
05/03/23 India's Contribution to Geometry 23
Sr no Publication Name of the book Author
1 . Motilal Banarasidass
Pvt.Ltd
Geometry in Ancient and Medieval India
T.A.Sarasvati Amma
2. ------ The Ganit Kaumudi of Narayana Pandit
Paramananda Singh
24
Thank You
Any Questions???
05/03/23