Brute force is an ugly but indispensable method in the math olympian’s toolbox. In the context of Euclidean geometry, brute force is better known as “side/angle/trigo-whacking”, depending on what sorts of quantities are being calculated. For example, you could label the 3 sides of a triangle as , and , then proceed to express every other possible side length or angle in terms of those 3 variables.

One useful theorem that can help the “whacking” process is Stewart’s Theorem, which allows us to express the length of a cevian (i.e. a line segment joining a vertex of the triangle to a point on the opposite side of the triangle). Stewart’s Theorem is easy to state:

**Stewart’s Theorem.** In the diagram above, the length of the cevian is given by the formula

I can never remember the equation above. Instead, I remember the proof in one line and derive the theorem!

**Proof: “Cosine rule for the angles at point .”**

Note that . By the cosine rule for , we have:

By the cosine rule for , we have:

Hence, we have the following:

**Done!**

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Cevian is a new term to me. Thank you for it.