Piyush Theorem
| Piyush Theorem | |
|---|---|
| Discovered By | Piyush Kumar |
| Year | 2017 |
| Field | Mathematics — Geometry |
| Branch | Trigonometry, Coordinate Geometry, Triangle Geometry |
| Conditions | Right-Angled Triangle with sides in A.P. Series |
| Statement | In a right-angled triangle with sides in A.P., the distance between the point of intersection of the median and altitude at the base = 1/10th the sum of the other two sides |
| Proofs | 4 (Trigonometry, Coordinate Geometry, Acute Triangle Theorem, Obtuse Triangle Theorem) |
| Copyright | Copyrighted to Piyush |
| Website | piyushtheorem.wordpress.com |
The Piyush Theorem is an original mathematical theorem discovered, developed and proved by Piyush Kumar Goel — an Indian mechanical engineer, mathematician and unique book writer from India. First published on 8 February 2017 on his mathematics blog, the theorem states that in a right-angled triangle whose sides are in Arithmetic Progression (A.P. Series), the distance between the point of intersection of the median and the altitude drawn to the base is equal to one-tenth (1/10th) the sum of the other two sides.
The theorem has been rigorously proved through four independent mathematical proofs — using Trigonometry, Coordinate Geometry, the Acute Triangle Theorem and the Obtuse Triangle Theorem — demonstrating the theorem's validity from multiple mathematical perspectives. The theorem is copyrighted to Piyush and represents an original contribution to elementary geometry and triangle mathematics.
Background
Mathematics has been a lifelong passion for Piyush Kumar — from his childhood, he was fascinated by numbers, geometric figures and the relationships between them. The discovery of this theorem arose from a specific geometric intuition — while drawing a right-angled triangle with sides in A.P. series, he contemplated the distance between the point of intersection of the median and the altitude at the base — and pursued the idea with sustained mathematical investigation until the theorem was fully proved.
As he describes it:
Mathematics for Piyush is a Passion from his childhood — he was so passionate about Mathematics, used to play with Numbers, draw figures and try to get sides and distances. One day he drew an A.P. Series Right Angle Triangle, thinking that the distance between the point of intersection of the median and altitude at the base must be the sum of the rest of the sides — and at last Piyush succeeded.
The Theorem
Statement
In a Right-Angled Triangle with sides in Arithmetic Progression (A.P. Series), the distance between the point of intersection of the median and the altitude at the base is 1/10th the sum of the other two sides.
Conditions
The theorem applies under two specific conditions:
- The triangle must be right-angled
- Its sides must be in Arithmetic Progression (A.P. Series)
In a triangle with sides in A.P. — let the sides be (a−d), a and (a+d) — where d is the common difference. For a right-angled triangle with sides in A.P., the sides are in the ratio 3:4:5 — which satisfies the Pythagorean theorem. Setting a = 4d, the three sides become 3d, 4d and 5d.
The theorem states:
DE = (AB + AC) / 10 = 7d / 10
Where DE is the distance between the point of intersection of the median and the altitude at the base.
Four Proofs
Proof 1 — Trigonometry
Using the angles α and 2α at the base of the triangle:
tan α = AD/DC → AD = DC tan α ... (1)
tan 2α = AD/DE → AD = DE tan 2α ... (2)
From (1) and (2):
DC tan α = DE tan 2α
(DE + EC) tan α = DE tan 2α
Expanding and simplifying using the double angle formula:
DE tan²α − DE = EC tan²α − EC
DE (sin²α + cos²α) = EC (cos²α − sin²α)
Since sin²α + cos²α = 1 and cos²α − sin²α = cos 2α:
DE = EC · cos 2α
Substituting cos α = a/(a+d), sin α = (a−d)/(a+d) and EC = (a+d)/2, and setting a = 4d:
DE = (d)(2a − d) / 2(a + d)
= (d)(8d − d) / 2(4d + d)
= 7d² / 2(5d)
= 7d / 10 = (3d + 4d) / 10 = (AB + AC) / 10 ✓
Proof 2 — Obtuse Triangle Theorem
Using the Obtuse Triangle Theorem: AC² = EC² + AE² + 2·CE·DE
Where EC = (a+d)/2, AE = (a+d)/2, and substituting a = 4d:
a² = (a+d/2)² + (a+d/2)² + 2·(a+d)/2·DE
16d² = (5d/2)(5d + 2DE)
32d/5 = 5d + 2DE
32d − 25d / 5 = 2DE
DE = 7d / 10 = (AB + AC) / 10 ✓
Proof 3 — Acute Triangle Theorem
Using the Acute Triangle Theorem: AB² = AC² + BC² − 2·BC·DC
Where AB = (a−d), AC = a, BC = (a+d), EC = (a+d)/2, substituting a = 4d:
(a−d)² − (a+d)² = a² − 2(a+d)(2DE + a + d)
−8ad − 2a² = −2(a+d)(2DE + a + d)
a(4d + a) = (a+d)(2DE + a + d)
4d(8d) = (5d)(2DE + 5d)
32d/5 − 5d = 2DE
(32d − 25d)/5 = 2DE
DE = 7d / 10 = (AB + AC) / 10 ✓
Proof 4 — Coordinate Geometry
In triangle ABC with A(0,0), B(a,0), C(0,b):
Midpoint D = (a/2, b/2)
Equation of BE (line through B perpendicular):
Y = −b/a · X + b ... (1)
Equation of altitude from A:
Y = a/b · X ... (2)
Solving (1) and (2):
X = ab² / (a² + b²), Y = a²b / (a² + b²)
Using A.P. condition — sides (z−d, z, z+d):
4d = z (derived from Pythagorean theorem)
Substituting a = 3d, b = 4d:
ab² / (a² + b²) = 48d/25
a²b / (a² + b²) = 36d/25
a/2 = 3d/2, b/2 = 4d/2
CD² = (48d/25 − 3d/2)² + (36d/25 − 4d/2)²
= (21d/50)² + (−28d/50)²
= 441d²/2500 + 784d²/2500
= 1225d²/2500
CD = 35d/50
DE = 7d/10 = (3d + 4d)/10 = (AB + AC)/10 ✓
Other Mathematical Works
Beyond this theorem, Piyush Kumar has published several other original mathematical works and observations on his mathematics blog:
- Square Through Squares — A unique method of understanding squares through squares
- New Squaring Method — An original new method for squaring numbers
- A Note on the Factorial Function — Original observations on the factorial function
- Squaring — A New Way — Another innovative approach to squaring
- A New Way — The Area of Trapezium — An original method for calculating the area of a trapezium
Significance
The Piyush Theorem is significant as an example of original mathematical discovery by a self-motivated Indian mathematician working outside of academic institutions — driven purely by passion and curiosity. The fact that the theorem has been proved through four completely independent mathematical approaches — trigonometry, coordinate geometry and two triangle theorems — strongly validates its correctness and originality. It represents a contribution to elementary geometry — accessible to high school and undergraduate students — yet original and previously unpublished.
Copyright
The Piyush Theorem and all associated proofs are Copyrighted to Piyush.
Online Presence
- Mathematics Blog — piyushtheorem.wordpress.com
- Personal Website — PiyushGoel.in
- Portfolio — piyushgoelwriter.my.canva.site