Using Geometry to Verify a CAD Solution
Waters, William M., Jr., Branoff, Theodore J., Clark, Aaron, Mathematics and Computer Education
Recently, one of the authors, who teaches engineering graphics (with a considerable CAD component), brought the following problem to the attention of the mathematics educator author:
You are given a sketch of a broken wheel from a wagon and asked to determine the size of a replacement wheel. The sketch includes the size of two chords and an angle on the wheel as shown in Figure 1. What is the diameter of the wheel? [1, p. 102].
The engineering graphics students were well aware of the method for locating the center, namely the intersection of the perpendicular bisectors of the two chords (See Figure 2) and that measurement of a scale drawing would enable them to determine the length of the diameter (The measurement of the included angle is required to determine the unique circle, since an infinite number of circles have two chords with a common endpoint and lengths of the given dimensions). The reading could be taken (to nine decimal places) directly from the computer, using the Computer Aided Design (CAD) software provided.
The students inquired about a mathematical approach to the problem. The first response involved the use of Heron's (or Hero's) formula and a little known theorem that the mathematics educator ran across while teaching a college geometry course. The original citation has been lost, but the problem now appears in the latest revision of Eves' text [2, p. 242] as problem 7.9(a).
2. STATEMENT AND PROOF OF THE THEOREM