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 x^{2} + y^{2}, substitute in the trigonometric expressions for x and y, and simplify as follows:
Now rearrange x^{2} + y^{2 }= 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