Intersecting CirclesDate: 05/13/99 at 17:36:45 From: Peter Subject: Intersecting circles problem Problem: Two circles intersect at A and B. A common tangent touches the two circles at S and T. Show that the line AB bisects the common tangent ST. I have tried various methods, e.g. using the intercept theorem and circle properties, and I am still stumped. I would appreciate a solution. Thank you. Date: 05/14/99 at 05:15:04 From: Doctor Floor Subject: Re: Intersecting circles problem Hi, Peter, Thanks for your question! We can use here that the power of a point with regard to a circle is constant. What is the power of a point w.r.t. a circle? Let P be a point, and let a line through P intersect the circle at points B and C. Then the power of P w.r.t. the circle is PB*PC. When P is inside the circle, the product is negative. For this problem we will consider P to be outside the circle. Why doesn't it matter what line through P intersects the circle? Let's look at a diagram: We can see that angle PDB = 180 deg - angle EDB = angle ECB = angle ECP, and that angle PBD = angle PEC, so triangles PDB and PCE are similar. Thus PD/PB = PC/PE, and PB*PC = PD*PE. We can conclude that the power is constant. When the line through P is tangent to a circle, say at point G, then of course the power of P w.r.t. the circle becomes PG*PG = PG^2. Now, for your problem let us consider a tangent common to two circles, and the line connecting the intersection points of these circles: The power of S w.r.t. the circle with center A is: SR^2 = SF*SE. The power of S w.r.t. the circle with center B is: SP^2 = SF*SE. So we have SR^2 = SP^2, and hence SR = SP, as required. If you have a math question again, please send it to Dr. Math! Best regards, - Doctor Floor, The Math Forum http://mathforum.org/dr.math/ Date: 05/14/99 at 22:52:12 From: Peter Ooi Subject: Re: Intersecting circles problem Thank you, Dr. Math, for your solution to the Intersecting Circles problem. My students will appreciate the solution. I knew the theorem ? or theory of power of a point w.r.t a circle was not specifically mentioned in the New South Wales HSC Maths syllabus but I can certainly make changes to the question to suit the syllabus. Thanks again and best regards from a teacher from Oz. |
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