The Apollonius' Theorem. Formula and Proof

• The medians in a triangle

• Definition: A median of a triangle is a line segment connecting a vertex of the triangle to the midpoint of the opposite side.
• Every triangle has three medians that meet at a single center point, called centroid.
• 

  The three medians of the triangle

• In the triangle △ABC m_a is the median connecting the vertex A to the middle of side BC, m_b is the median connecting the vertex B to the middle of side AC and m_c is the median connecting the vertex C to the middle of the side AB. All the medians intersect at point G, the centroid.
• The point G divides the lengths of the three medians as follows:

• ^AG/_AM = ^BG/_BP = ^CG/_CN = ²/₃
• ^GM/_AM = ^GP/_BP = ^GN/_CN = ¹/₃
• ^GM/_AG = ^GP/_BG = ^GN/_CG = ¹/₂
  

• Apollonius' Theorem

• In a triangle △ABC, for which we know the lengths of the sides "a", "b" and "c", having the opposite vertices A, B and C, respectively, there is a relationship between the lengths of the sides and the lengths of the medians m_a, m_b and m_c, where m_a is drawn from vertex A to side "a", m_b is drawn from vertex B to side "b", and m_c is drawn from vertex C to side "c":

• b² + c² = 2 × (m_a² + (^a/₂)²) =>
• m_a² = ^{(b2 + c2)}/₂ - (^a/₂)²
• a² + c² = 2 × (m_b² + (^b/₂)²) =>
• m_b² = ^{(a2 + c2)}/₂ - (^b/₂)²
• a² + b² = 2 × (m_c² + (^c/₂)²) =>
• m_c² = ^{(a2 + b2)}/₂ - (^c/₂)²

• Proof of the Apollonius' Theorem

• We prove Apollonius' Theorem using the Law of Cosines, also called the Generalized Pythagorean Theorem, which is also used to calculate the angles' measures in a triangle.
• » Law of Cosines or the Generalized Pythagorean Theorem. Formula and Proof
• 

  The three medians of the triangle Proof of the Apollonius' Theorem

• Let median AM, of lenght m_a, divide the side BC, of length "a", into two equal parts, of length BM = CM = ^a/₂. Also the legths of AB = c, AC = b, BC = a.
• Let the measure of the angle ∡AMB = θ. Since two angles on a straight line add up to 180° then the adjacent angle ∡AMC = 180° - θ.

• => cos(180° - θ) = - cos(θ)
• » Cosine and Arccosine in a Right Triangle

Apply the Law of Cosines to the triangle △ABM, making sure to also include the angle ∡AMB = θ in the formula:

• c² = m_a² + (^a/₂)² - 2 × m_a × ^a/₂ × cos(θ)
• => [1] c² = m_a² + (^a/₂)² - m_a × a × cos(θ)

Apply the Law of Cosines to the triangle △ACM, making sure to also include the angle ∡AMC = 180° - θ in the formula:

• b² = m_a² + (^a/₂)² - 2 × m_a × ^a/₂ × cos(180° - θ)
• Substitute for the expression: cos(180° - θ) = - cos(θ)
• The initial above relation becomes:
• [2] b² = m_a² + (^a/₂)² + m_a × a × cos(θ)

• Add the two equations [1] and [2] together. The terms "m_a × a × cos(θ)" and "- m_a × a × cos(θ)" cancel each other:

b² + c² = 2 × (m_a² + (^a/₂)²) - Apollonius' Theorem

• Calculation example

• Let's calculate the medians m_a, m_b and m_c of a triangle if all the sides' lengths are given: a = 12, b = 10 and c = 8.
• Apply the Apollonius' Theorem and solve for the medians: