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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:
𝑍2 = 𝑋2 + π‘Œ2
𝑋 2 = 𝐴𝐢 2 + 𝐡𝐢 2 𝑋 2 = 𝐴𝐢 2 +
𝑍
π‘Œ
– 𝑋+
2
10
π‘Œ 2 = 𝐴𝐢 2 + 𝐢𝐸 2 π‘Œ 2 = 𝐴𝐢 2 +
𝑍
π‘Œ
+ 𝑋+
2
10
𝑋 2 – π‘Œ 2 = 𝐴𝐢 2 – 𝐴𝐢 2 +
𝑍
π‘Œ 𝑍
π‘Œ
−𝑋+
+ – 𝑋+
2
10 2
10
𝑋+π‘Œ 𝑋– π‘Œ = 𝑍
𝑋−π‘Œ = −
Copyrighted © Piyush Goel (1987)
−𝑋+
𝑍
π‘Œ 𝑍
π‘Œ
– 𝑋 + – –𝑋 +
2
10 2
10
π‘Œ
5
𝑍
5
® Geometry (Euclidean/Algebraic)
2
𝑍 = 5π‘Œ − 5𝑋 … 𝑃𝑒𝑑 π‘‘β„Žπ‘–π‘  π‘£π‘Žπ‘™π‘’π‘’ π‘–π‘›π‘‘π‘œ 𝑍 2 = 𝑋 2 + π‘Œ 2
5π‘Œ − 5𝑋
2
= 𝑋2 + π‘Œ2
25π‘Œ 2 + 25 𝑋 2 – 50π‘‹π‘Œ = 𝑋 2 + 𝑦 2
24𝑋 2 – 50π‘‹π‘Œ + 24π‘Œ 2 = 0
24𝑋 2 – 32π‘‹π‘Œ – 18π‘‹π‘Œ + 24π‘Œ 2 = 0
8𝑋 3𝑋 – 4π‘Œ – 6π‘Œ 3𝑋 – 4π‘Œ = 0
3𝑋 − 4𝑦 8𝑋 – 6π‘Œ = 0
2 3𝑋 − 4π‘Œ 4𝑋 – 3π‘Œ = 0
3𝑋 − 4π‘Œ 4𝑋 – 3π‘Œ = 0
3𝑋 − 4π‘Œ = 0
4𝑋 – 3π‘Œ = 0
3𝑋 = 4π‘Œ
πŸ’π‘Ώ = πŸ‘π’€
3 π‘Œ
4π‘Œ
3𝑋
= π‘œπ‘Ÿ 𝑋 =
π‘œπ‘Ÿ π‘Œ =
4 𝑋
3
4
πŸ’ 𝒀
πŸ‘π’€
πŸ’π‘Ώ
= 𝒐𝒓 𝑿 =
𝑢𝑹 𝒀 =
πŸ‘ 𝑿
πŸ’
πŸ‘
𝑃𝑒𝑑 π‘£π‘Žπ‘™π‘’π‘’ π‘œπ‘“ π‘Œ 𝑖𝑛 π‘’π‘žπ‘›. 5π‘Œ − 5𝑋 = 𝑍
5∗
4𝑋
– 5𝑋 = 𝑍
3
20𝑋 – 15𝑋 = 3𝑍
5𝑋 = 3𝑍
𝑋 3
=
𝑍 5
π‘Œ 4
=
𝑋 3
π‘†π‘œ 𝑋 = 3; π‘Œ = 4; 𝑍 = 5 π‘‘β„Žπ‘Žπ‘‘ 𝑖𝑠 π‘Žπ‘› 𝐴 𝑃 π‘†π‘’π‘Ÿπ‘–π‘’π‘ .
Copyrighted © Piyush Goel (1987)
® Geometry (Euclidean/Algebraic)
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