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.
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.
Using triangle ACB, it follows that
Triangle AEC is a right triangle and x = AE while y = CE. See 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.
Alternative proof using similar triangles.
In Figure 10, triangles AEC and BEC are similar to triangle ACB, so triangle AEC is similar to triangle BEC. Thus we have that
which can be expressed as
Simplifying this expression gives
Complete the square as shown previously to determine the equation of a circle containing the dots (x,y).
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