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My first construction: equilateral triangle

My first construction: equilateral triangle

Author: Christopher Danielson

To introduce geometric constructions with compass and straightedge from the perspective of why they work, rather than as a series of steps.

This packet demonstrates the process of constructing an equilateral triangle, and establishes a model for understanding why constructions work.

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Geometric constructions are fascinating. There is something pure and elegant about using only a compass and an unmarked straightedge to draw beautiful, complicated and sophisticated images.

But constructions also offer a window on a rich world of mathematics. They can help us explore geometric relationships, reason geometrically and develop a better understanding of proof in mathematics.

This is the first in a series of packets intended to focus on why constructions work, rather than solely on their mechanics. To be sure, there are steps to follow. But at every turn these packets will focus on the meaning of these steps.

In order to appreciate this first construction, it is helpful to know something about circles. So if you haven't done so already, read my packet on circles. Then get your compass, your straightedge and some plain white paper; let's construct an equilateral triangle.

The construction

This video demonstrates using a compass and straightedge to construct an equilateral triangle.

NOTE: Please forgive the squashed appearance of the video. The aspect ratio of my document camera does not match the aspect ratio of Sophia exactly, so the video becomes compacted vertically-my circles look squashed and my vertical lengths look shorter than my horizontal lengths. I will work on a fix for this in future videos.

One more time: a smaller equilateral triangle

This video demonstrates a useful shortcut and walks quickly through the construction a second time to draw an equilateral triangle of a different size.

What stays the same? What changes?

This video compares the two constructions and asks what stays the same and what changes when we do the construction a second time.

Why does it work?

As we work with constructions, it will be important to ask "Why does this construction work?" and it will be important to be skeptical about any explanations that are offered. Don't let me bully into believing that two angles are the same size as each other-demand explanations. You can do this rhetorically by asking these questions of yourself. And you can do it quite literally by using the Q&A feature of Sophia packets to ask questions.

To recap, this construction works because we locate point C that is the same distance from A as it is from B, and this distance is the same as the distance between A and B. We are using the definition of circles and the mathematical properties of a compass to ensure these distances are the same as each other.