Strategies in problem solving

Introduction to Mathematical Proofs

Info

A solution should be relatively short, understandable, and hopefully have a touch of elegance. It should also be fun to discover. Transforming a nice, short little geometry question into a ravening monster of an equation by textbook coordinate geometry does not have the same taste of victory as a two-line vector solution

Example

Show that the perpendicular bisectors of a triangle are concurrent.

Proof. Call the triangle $ABC$. Now let $P$ be the intersection of the perpendicular bisectors of $AB$ and $AC$. Because $P$ is on the $AB$ bisector, $|AP|=|PB|$. Because $P$ is on the $AC$ bisector, $|AP|=|PC|$. Combining the two, $|BP|=|PC|$. But this means that $P$ has to be on the $BC$ bisector. Hence all three bisectors are concurrent.

(Incidentally, $P$ is the circumcentre of $△ABC$.) $\square$

Understand the problem

• “Show that…” or “Evaluate…” questions in which a certain statement has to be proven true, or a certain expression has to be worked out
• “Find a…” or “Find all…” questions, which requires one to find something (or everything) that satisfies certain requirements
• “Is there a…” questions, which either require you to prove a statement or provide a counterexample (and thus is one of the previous two types of problem)

Note

Not all problems fall into these neat categories

Understand the Data i