Working on the perfect–triangle problem was a real mountaintop experience in mathematics. Our problem–solving path started with basic geometry, followed by a little algebra, then more geometry and more algebra. Our tour was enhanced by using a computer not to solve the problem but to explore its solution.

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Central to most problem-solving strategies is the desire to state and solve a problem in its most general form. Unfortunately, often the most general form of a problem is not easy to state. Moreover, the proof of the generalization is often beyond the abilities of many of our students. So like the idea of “synthesis” in test construction, generalizing often becomes that part of the problem-solving process that is admired from afar but seldom used.

How a geometric insight can add to the richness of our students' experiences when they first encounter the solutions to two equations in two unknowns. Includes sample graphs.