Piyush Theorem 3

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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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