Tangent Circle ConstructionDate: 12/02/96 at 23:48:58 From: Jimmy Midland Subject: Euclidean geometry Given a circle and two points inside of the circle, construct another circle which passes through the given points and is tangent to the given circle. Date: 12/05/96 at 10:13:10 From: Doctor Yiu Subject: Re: Euclidean geometry Dear Jimmy, This is one of the famous problems in geometric construction. Here is one construction (without proof). If you would like an explanation of the basic principle behind the construction, please write back. I: First consider the case when A and B are EQUIDISTANT from the center O of the given circle. Draw the perpendicular bisector of AB to meet the given circle at two point P and Q. The two circles ABP and ABQ are each tangent to the given circle. II: Now, suppose that A and B are NOT equidistant from the center O of the given circle. (1) Draw the perpendicular bisector of AB to meet the given circle at a point P. (There are two such intersections. Choose either of them). (2) With P as the center, draw a circle through A (it also passes through B), and mark its intersections with the given circle as C and D. (3) The lines AB and CD meet at a point K (since A and B are not equidistant from the center of the given circle). (4) Suppose A is BETWEEN K and B. (If not, simply change the labels of the points A and B). Draw a circle with KB as DIAMETER, and then through A draw the perpendicular to KB to meet this circle at X. (5) Draw the circle with center K passing through the point X. Mark the intersections of this circle with the GIVEN circle as M and N. (6) Mark the intersection of OM and the perpendicular bisector of AB. With this as center, draw a circle passing through M. This circle is tangent to the given circle and passes through the given points A and B. (7) Repeat (6) with the line ON (instead of OM). This gives a SECOND circle meeting the same requirement. Here is a diagram illustrating these steps: and a sketch made with The Geometer's Sketchpad is at: http://mathforum.org/dr.math/sketches/tangent_circle.gsp To open the sketch, you'll need at least a demo version of the Geometer's Sketchpad, which you can obtain at: http://mathforum.org/dynamic.html -Doctor Yiu, The Math Forum Check out our web site! http://mathforum.org/dr.math |
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