Expert Answer
Stewart's Theorem is a geometric theorem that relates the lengths of the sides of a triangle to the lengths of the segments formed by the medians of the triangle. Let's consider a triangle ABC with sides of lengths a, b, and c, and medians AD, BE, and CF, where D, E, and F are the midpoints of the sides BC, AC, and AB, respectively.Stewart's Theorem states that in a triangle, the square of the length of a median is equal to the sum of the squares of half the length of the corresponding side and the lengths of the two segments into which the median divides that side. Mathematically, it can be expressed as:m² = (1/4)(2b² + 2c²) - (1/2)a²where m represents the length of the median corresponding to side a.To prove Stewart's Theorem, we can use the Law of Cosines and the Pythagorean Theorem. Let's derive the formula for one of the medians, say AD.Using the Law of Cosines, we have:b² = c² + a² - 2ac*cos(B)Since D is the midpoint of BC, we have BD = DC = a/2. Applying the Pythagorean Theorem to triangle ABD, we get: