Stewart's Theorem - Explanation and Proof - Testbook.com

Understanding Stewart’s Theorem

In geometry, Stewart’s theorem establishes a correlation between the lengths of a triangle's sides and the length of its cevian. This theorem is credited to Scottish mathematician Matthew Stewart, who introduced it in 1746.

Statement – Assuming a, b, c are the lengths of a triangle ABC, and d is the length of a cevian dividing the side of length 'a' into two segments of lengths m and n, where m is adjacent to side c and n is adjacent to side b, then the following relationship holds true-

Proof:

The proof of this theorem can be established using the law of cosines.

Let's denote the angle between side m and side 'd' as θ, and the angle between side n and side 'd' as θ′. θ′ is the sum of θ and cos θ′ is (−cos θ). The law of cosines for angles θ′ and θ is as follows-

 

Multiply the first equation by n, the second equation by m, and then add them to eliminate cos θ, we get-

 

Thus, we have the required proof.

Stewart’s Theorem can also be proven by drawing a perpendicular from the triangle's vertex to the base and using the Pythagorean theorem to express the distances b, d, c, in terms of altitude. Both sides of the equation can be algebraically reduced to the same expression.

Stewart’s Theorem is highly important due to its applicability in solving various geometric problems.