Apollonius Theorem is a popular part of elementary Geometry that is related to the length of the median of a triangle and length of its sides too. There are different names for the theorem is different regions and it can be proved by the Pythagorean Theorem by using cosine rule as well as vectors too. The theorem was named after the name of a Greek mathematician i.e. Apollonius of Perga.

Theorems are the popular statements in mathematics that had proven results based on previously set statements like axioms. The theorems are the proven results with accurate assumptions and a set of different axioms. When you will study mathematics during your school time or college periods, Theorems are an important part that is asked in the form of questions.

Medians are the crucial set of components in geometry that are closely related to the Triangle and is independent of the geometric shapes too. In the case of Apollonius’ Theorem, you know the relationship among medians and sides of a triangle. This is a popular theorem that connects the medians of a triangle with sides of a Triangle.

Based on the Apollonius’ Theorem, the sum of the squares of any two sides would be equal to the twice of the square of the third side that bisects the third side too.

If O is the midpoint of MN, one of the sides of the triangle (LMN), then prove that

LN² + LM² = 2 {MO² + LO²}.

Generally understanding the proofs of Theorems are difficult. With the proper knowledge and educational background, you could derive the proof of a Theorem quickly. There is no quick way or shortcut method to construct the proof of the theorem. You should have a strong understanding of the concept and multiple logics to devise your proof.

By continuous practice, you would be able to cultivate the proof the theorem by your own. Here, are few steps to help you how can you write the proof for a tough theorem like the Apollonius’ Theorem.

First, you should identify the problem what are you trying to prove and what is the final statement. You should also define the assumptions and questions that are necessary to work on before you start with the proof.

When you are trying to understand the tough mathematical problems then worm on diagrams so that proof can be visualized clearly in front of your eyes. Label each side and angle properly and get an idea of medians as well. You could refer books or online websites for the same to help you out.

This is a common thing getting stuck on the proof and getting confused how can you devise the solution of a problem. The best idea is to make a list of questions and try to solve them one by one to find the solution. This is always better to ask questions and experiment multiple time before you blindly cram the proof.