1267. Apollonius' theorem

    Classical Geometry: unparalleled beauty...

    Apollonius' theorem states that the mean of the areas of the squares on any two sides of any triangle equals the sum of the area of the square on half the third side and the area of the square of the median bisecting the third side... See below:

 

    Pepe Chapuzas proved the theorem:

    Dear Teacher:
    The cosine law states that

a²  =  m² + c²/4 − m · c · cos θ

b²  =  m² + c²/4 − m · c · cos θ'  =  m² + c²/4 + m · c · cos θ

(θ and θ' are suplementary)

    So, the mean of  a²  and  b²  is

(a² + b²) / 2  =  m² + c²/4

    Mr. López, when the triangle is isosceles, Apollonius' theorem becomes Pithagoras' theorem, doesn't it?