Piyush Theorem 2

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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)
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