1. TWO SIMPLE PROOFS OF
PYTHAGORAS THEOREM
Presented by :
Dr. Anant W. Vyawahare,
Nagpur.
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2. • These proofs of Pythagoras theorem are given
by Ganesh Daiwadnya.
• Ganesh Daiwadnya, an astronomer and a
mathematician, was born in 1507 ad. at a
place Nandgaon, in Maharashtra, 40 kms .
South of Mumbai.
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3. The texts written by Ganesh Daiwaidnya ,all in
Marathi language, are:
• Laghu & Bruhat Tithi Chintamani,
• Buddhi-Vilasini – a commentary on Bhaskara`s
Lilavati,
• A commentary on Bhaskara`s Sidhanta
Shiromani
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4. • Vrindavana- Tika- a text containing
elementary puzzles,
• Graha-Laghava- a treatise on astronomy,
( His father, Keshava, an observational
astronomer, wrote a book ,Graha Kautaka.
was ,then, the only text available in Marathi
language)
• Shradha Nirnay, Parva Nirnay, etc.
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5. • The proofs given below are from
Buddhivilasini – a commentary on Bhaskara`s
Lilavati ( 1545 ad.)
• Of all the proofs available, these proofs seem
to be simple.
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6. • First proof:
• Consider a right angled triangle ABC, right angled at A with base BC.
• To prove, (BC)2 = (AB)2 + (AC)2
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C
A B

7. • Draw a perpendicular AD from A to meet BC in
D.
• Consider three right angle triangles ABD, ADC,
and ABC.
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C
A B
D

8. • Three triangles are similar because
• (i) a right angle is common in all the
three triangles,
• (ii) angle B is same in triangles ABD and
ABC,
• (iii) angle C is same in triangle ADC and
ABC .
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9. • Now, triangles ABD Ξ ABC,
• Hence, BD /AB = AB / BC,
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C
A B
D
Therefore, BD = (AB)2 / BC, ……………… (1)

10. • Similarly, triangles ADC ≡ ABC,
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This gives, DC / AC = AC / BC
Hence, DC = (AC)2 / BC ……………………(2)
C
A B
D

11. • Add (1) and (2)
we get, BD +DC = (AB)2 / BC + (AC)2 / BC
• That is, BC = (AB)2 / BC + (AC)2 / BC,
Or, (BC)2 = (AB)2 + (AC)2
• This completes the proof.
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12. • Another proof:
In triangles ABC and ACD,
Cos θ = AC/BC = CD/AC,
This gives AC2 = BC.CD,…………………………1
Similarly, In triangles ABC and ABD,
Sin θ = AB/BC = BD/AB,
This gives AB2 = BC.BD,…………………………2
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13. • Add (1) and (2)
AB2 + AC2 = BC(BD + CD),
=BC.BC
=BC2,
This completes another proof.
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14. • Ganesh Daiwaidnya also invented two
Pythagorean triplets:
(m2 – n2, 2mn, m2 + n2 ),
and (p2 – q2, 2pq , p2 + q2 ),
and invented a new triplet
[(m2 – n2)(p2 – q2) , 2mn (p2 – q2),
(m2 + n2)(p2 – q2), ]
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15. References:
• T.S. Bhanu Murthy: A modern introduction to
ancient Indian Mathematics.( Wiley Eastern
Publ. , Singapore, 1992)
• T.A.Saraswati Amma: Geometry in ancient and
medieval India,( Motilal Banarasi Das publ.,
New Delhi, 1999)
• Dr. S. Bhalachandra Rao: Indian Mathematics
and Astronomy,(Jnana Deep Publications,
Banglore, 1994)
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16. THANK YOU
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