Objective: To prove that the dots (x,y) at the corner of the carpenter's square obtained from the procedure depicted in Figures 2 - 6 lie on a circle and find a parametric representation of the circle.

Procedure: Using trigonometry determine the equation of a circle that contains the points (x,y).

In Figure 7, label angle BAC as . Note that the dashed line from vertex C is a perpendicular to segment AB. This gives us two other right triangles AEC and BEC. We show this in Figure 10.

Figure 10.

Using triangle ACB, it follows that

Triangle AEC is a right triangle and x =  AE while y = CE. See Figure 11.

Figure 11.

Using triangle AEC we get

From Equation (1) we can substitute for AB to get

To show that (x, y) lies on a circle we start with the expression x2 + y2, substitute in the trigonometric expressions for x and y, and simplify as follows:

Now rearrange x2 + y2 = dx and complete the square; we get

This final equation represents a circle centered at (d/2, 0) with radius d/2.

The expressions in (2) provide a parametric representation for the circle containing the dots generated by the carpenter's method. We have,

By varying parameter from 0 to a semicircle is generated. For a full circle vary parameter from 0 to .

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DRH 12/24/2004