Draft:Elwasif's Proof

---

The median of a trapezoid is a segment that joins the midpoints of the non-parallel sides. This article presents a proof of the Median of the Trapezoid Theorem, developed by Moustafa Elwasif, which states that the median is half the sum of the lengths of the trapezoid's bases. This theorem has practical significance in geometry for its simple yet powerful relationship.

Background:

In geometry, a trapezoid (or trapezium) is a quadrilateral with at least one pair of parallel sides called bases. The median, also known as the midsegment, is a crucial element in the study of trapezoids due to its unique properties.

Statement of the Proof:

The Median of the Trapezoid Theorem asserts:

$m=\frac{(a+b)}{2}$

​ where

$m$

is the length of the median, and

$a$

and

$b$

are the lengths of the bases.

• File:Trapezoid with minimum labeling.png

Methodology:

The proof involves defining a variable $k$ that represents the difference between the median and either base $a$ or $b$. The relationships are expressed as: $k=(b-m),(m-a),(\frac{b-a}{2})$

From these relationships, the expressions for $a$, $b$, and $m$ are derived:

$b=(m+k),(a+2k)$

$a=(m-k),(b-2k)$

$m=(b-k),(a+k)$

• File:Elwasif's Proof of Median of The Trapezoid Theory Diagram with labeling.png

Verification:

By substituting $m=\frac{a+b}{2}$​ into the expressions for $k$ and simplifying, the equality of these relationships is confirmed, thus proving the theorem.

Authorship and Development:

The Proof for the Median of the Trapezoid Theorem was developed by Moustafa Elwasif at the age of 15, in 2016. The idea originated during a geometry test that was created and graded by Prof. Mohammad Jarkas. He pointed out that the solution Moustafa wrote on the test paper was achieved through a method that was unheard of and said after a brief examination that it could be a potential proof.

During March 2023, Dr. Prof. Samuel Moveh agreed to examine this proof, then and confirmed its validity in November 2023, suggesting that it should be published.

Significance and Impact:

This theorem is significant in the study of trapezoids as it provides a simple yet powerful relationship between the bases and the median. It is widely used in various geometrical applications and problems. The theorem's simplicity makes it a useful tool in various geometrical applications, particularly in solving problems related to trapezoidal shapes in both academic and practical contexts.