Geometry: An Interactive Journey to Mastery

Exploring Geometric Constructions

Lesson 26

Topics
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x Constructible numbers.
x Constructible regular polygons.
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ZLWKQRPDUNLQJV௘ͽDQGDFRPSDVVDVWRROVLWLVSRVVLEOHWRGUDZDOLQHVHJPHQWRIOHQJWKa if given
only a line segment of length 1 already drawn on the page.
Summary
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compass? This classic question from the ancient Greek scholars has proved to be surprisingly rich and complex
in its mathematics. In this lesson, we explore the basic constructions we can perform with these primitive tools
and survey the mathematics of constructible numbers and constructible regular polygons.
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Given a line segment AB drawn on a page, show how to construct an equilateral triangle with AB as its base.
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6HWWKHFRPSDVVVRWKDWLWVOHQJWKLVAB, and draw two circles of this
radius, one with endpoint A as center and one with endpoint B as center.
Let C be one of the points of intersection of the two circles.
Then, AC is a radius of the circle with center A, so AC = AB.
By similar reasoning, BC = AB. Thus, triangle ABC is equilateral.
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We can complete the picture by drawing the line segments
AC and BC.

AB

C

Figure 26.1