Isosceles Right Angled Triangle

1. Isosceles Right Angled Triangle

2. 1 1 The three sides cannot be all integers

3. 5 4 3 Find a right angled triangle with integralsides near to be isosceles such as

4. Question • The difference between two legs is 1 • What is the next triangle? • Are there sequences of the legs with ratio tending to 1 : 1 ?

5. a + k a + 1 a The three sides should be a, a + 1 and a + k where a, k are positive integers and k > 1

6. Relation between a and k • a2 + (a + 1)2 = (a + k)2 • a2 – 2(k – 1)a – (k2 – 1) = 0 • as a > 0

7. Discriminant • 2k(k – 1) is a perfect square • For k, k – 1, one is odd, one is even • Since k, k – 1 are relatively prime, odd one = x2, even one = 2y2 • Then x2 – 2y2 = 1 • 2k(k – 1) = 4x2y2

8. Find the three sides

9. Find the three sides If k is odd, then k = x2, then the three sides are: 2y2 + 2xy, x2 + 2xy, x2 + 2xy + 2y2 If k is even, then k = 2y2, then the three sides are: x2 + 2xy, 2y2 + 2xy, x2 + 2xy + 2y2 Both cases have the same form.

10. Pell’s Equation • x2 – 2y2 = 1 • How to solve it? • Making use the expansion of into infinite continued fraction

11. Continued Fraction

12. Sequence of the Fractions

13. Sequence of the Fractions {fn} is generated by the sequence{1, 2, 2, 2, …. } where h1 = 1, h2 = 3, k1 = 1, k2 = 2 hn = hn-2 + 2hn-1, kn = kn-2 + 2kn-1 for n > 2

14. Solution of Pell’s Equation x2 – 2y2 = 1 • (hn, kn) is a solution of x2 – 2y2 = 1 when n is even. • (hn, kn) is a solution of x2 – 2y2 = -1 when n is odd. • This is the full solution set.

15. Sequences of the Sides

16. Properties of Sequences The triangle tends to be isosceles