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: