Creating Tangents to CirclesDate: 10/24/2007 at 09:53:53 From: Princy Subject: Drawing tangents to circles I know how to draw a tangent from a point (outside a circle) to the circle. How can I create 4 distinct tangents common to 2 circles (d > a + b)? Date: 10/24/2007 at 15:19:07 From: Doctor Wilkinson Subject: Re: Drawing tangents to circles Thanks for the nice problem, Princy! Let's start with a picture. You have a circle with center A and radius a and a circle with center B and radius b, and the distance d between A and B is greater than a + b, so that the circles don't intersect. Draw the line AB. The circles may lie on opposite sides of a common tangent line or they may lie on the same side, and there are two such tangents in each case. Let's look at the second case, and let E be the point of tangency on the circle with center A and F be the point of tangency on the circle with center B. Then EF intersects AB at some point X which is not between A and B (the case where a = b and EF is parallel to AB is easy). Since you know how to draw a tangent to a circle from a point outside the circle, the problem will be solved if you can figure out how to find the point X. Suppose a > b, so that B is between A and E. Then you have a big right triangle XAE and a smaller right triangle XBF, and these triangles are similar since they share the angle AXE. You also know the side AE = a and the side BF = b and the distance d = AB. Now maybe you can fool around with proportions to see how to construct the segment BX. The other case can be solved in much the same way. Write back if you still need help with any of this and tell me what you've tried. - Doctor Wilkinson, The Math Forum http://mathforum.org/dr.math/ |
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