1 Piyush Theorem #3 Statement: In a Right-angled Triangle, if distance between middle point and perpendicular point and sum of other two sides is divisible by 10, then sides will be in A.P. Series. Let’s prove it: 2 = 2 + 2 2 = 2 + 2 2 = 2 + – + 2 10 2 = 2 + 2 2 = 2 + + + 2 10 2 – 2 = 2 – 2 + −+ + – + 2 10 2 10 + – = − = − Copyrighted © Piyush Goel (1987) −+ – + – – + 2 10 2 10 5 5 ® Geometry (Euclidean/Algebraic) 2 = 5 − 5 … ℎ 2 = 2 + 2 5 − 5 2 = 2 + 2 25 2 + 25 2 – 50 = 2 + 2 24 2 – 50 + 24 2 = 0 24 2 – 32 – 18 + 24 2 = 0 8 3 – 4 – 6 3 – 4 = 0 3 − 4 8 – 6 = 0 2 3 − 4 4 – 3 = 0 3 − 4 4 – 3 = 0 3 − 4 = 0 4 – 3 = 0 3 = 4 = 3 4 3 = = = 4 3 4 = = = . 5 − 5 = 5∗ 4 – 5 = 3 20 – 15 = 3 5 = 3 3 = 5 4 = 3 = 3; = 4; = 5 ℎ . Copyrighted © Piyush Goel (1987) ® Geometry (Euclidean/Algebraic)

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# Piyush Theorem 3