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1 Piyush Theorem #2 Statement: In an AP Series △ Triangle, if distance of median point to perpendicular point and sum of other two sides is divisible by 10, then triangle is a Right Angle Triangle. What we have: Sides are in AP Series (a-d, a, a+d) Let’s prove it: = ( + )/10 10. = ( − ) + 5. + 5. = 2 − 5( − ) + 5( − ) = 2 − 5 – 5 + 5 – 5 = 2 − 5 – 5 = 2 − — – ( = ) 5( − ) = 2 − Copyrighted © Piyush Goel (1987) − − . 1 ® Geometry (Euclidean/Algebraic) 2 , ^2 = ^2 – ^2 ^2 = ^2 – ^2 ^2 − ^2 = ^2 – ^2 ( + )( − ) = ( + ) ( – ) ( + )( – ) = ( + – ) ( − + ) ( + )( − ) = (2 − ) – = (2 − ) ( + ) . 1 5(2 – ) / ( + ) = (2 − ) 5/ ( + ) = 1 5 = + 5 – = = 4 ℎ , ( + )^2 = ^2 + ( − ) ^2 ( + )^2 – ( − ) ^2 = ^2 ( + + – )( + – + ) = ^2 (2)(2) = ^2 = 4 . Copyrighted © Piyush Goel (1987) ® Geometry (Euclidean/Algebraic)