Arbelos ConstructionDate: Fri, 10 Mar 2000 13:26:51 -0500 (EST) From: Mark Yates Subject: Pappus' ancient theorem I have made many compass sketches and Geometer's Sketchpad sketches, alas to no avail. My question is this: Is there a Euclidean construction for the circles that get sandwiched in the Arbelos? I reference page 133 in _A Survey of Geometry_ by Howard Eves, copyright 1972. Picture segment AC. Draw a semicircle on it. Now place an arbitrary point B on AC. Draw semicircles BC and AB. Can a circle be constructed tangent to all three semicircles using Euclidean tools? I have narrowed my center to the semicircle whose center is halfway between the midpoint of the smaller semicircle and the endpoint of segment AC. For a picture, see circle C4 in theorem 4 at The Arbelos by Dr Peter Woo http://www.biola.edu/academics/undergrad/math/woopy/arbelos.htm Thanks! Date: 10/12/2001 at 07:24:25 From: Doctor Floor Subject: Re: Pappus' ancient theorem Hi, Mark, Thanks for sending your interesting question to Ask Dr. Math. The arbelos, the shoemaker's knife, is a very beautiful chapter in Euclidean geometry, in my opinion. The construction of the "incircle" of the arbelos is shown in the following figure I made for you: Let me give the description: Let J and M be the points on the smaller semicircles, such that JD and ME are perpendicular to AB. The circle with center J through A and B meets the arbelos semicircles at two more points, Q and N. The circle with center M through B and C meets the arbelos semicircles again at N and also at point P. The circle through P, Q, and N is the arbelos' incircle. Its center can be found by intersecting DP, EQ, and the line through N and the center of the largest semicircle. For the construction one of the red circles can be omitted. If you need more help, just write back. Best regards, - Doctor Floor, The Math Forum http://mathforum.org/dr.math/ |
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