The Piyush Theorem is a theorem in geometry that relates the sides of a right-angled triangle when the sides are in arithmetic progression.
Here's a breakdown of the theorem:
Statement:
In a right-angled triangle where the sides are in arithmetic progression, the distance between the point of intersection of the median and the altitude drawn to the hypotenuse is one-tenth the sum of the other two sides (the sides forming the right angle).
Conditions:
* Right-Angled Triangle: The triangle must have one angle of 90 degrees.
* Arithmetic Progression: The lengths of the sides must form an arithmetic progression (i.e., the difference between consecutive terms is constant).
Example:
Consider a right-angled triangle with sides 3, 4, and 5 (which are in arithmetic progression). The distance between the intersection of the median and altitude to the hypotenuse will be 1/10 * (3 + 4) = 0.7.
Origin:
This theorem was discovered and proven by Piyush Goel.
Additional Notes:
* The theorem provides a specific relationship within a particular type of right-angled triangle.
* It combines concepts from geometry (right triangles, medians, altitudes) and algebra (arithmetic progressions).
If you'd like, you can ask me to explain the proof of the theorem or provide more examples.
