Hello, I hope you all had a great day! In this post, we will be exploring the Apollonius Theorem, which can be thought of as a special case of the Stewart Theorem. The Apollonius Theorem will relate the length of the median of the triangle to the length of its sides. The Apollonius Theorem states that the sum of the squares of any two sides of any triangle equals twice the square on half the third side, together with twice the square on the median bisecting the third side. So, in other words, c^2+b^2=2(m^2+d^2) in the diagram below. Like the Stewart Theorem, we will prove this theorem using the law of cosines.
<Proof>
b^2=m^2+d^2+2dmcos𝜃 ….①
c^2=m^2+d^2+2dmcos𝜃’ (<= cos𝜃=cos𝜃’ since 𝜃 and 𝜃’ are supplements)
=m^2+d^2 - 2dmcos𝜃 ….②
① + ② yields
b^2+c^2=2(m^2+n^2), which is the Apollonius Theorem.
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