4. THREE INTERSECTING
CIRCLES
The intersection points of two intersecting circles determine a straight line. What happens if we have three intersecting circles?
Exploring We need first to examine an interesting property of the circle. Let P be a point external to a given circle, and let a line through PQR meet the circle in points Q, R. Also, let PT be tangent to the circle at T as in the diagram. Reproduce a copy of this diagram with a larger scale, and for different lines PQR, measure PQ, PR. Tabulate your results as below. What appears to be true?

Power of a Point The evidence points to the result: PQ.PR = PT ^{2}. 
It is easy to see why these two angles must be equal. 
Consider two circles intersecting in points A, B. The line AB is called the radical axis of the two circles. It has a special property.
Let P be an arbitrary point on AB. What can you say about PQ . PR? What can you say about PQ' . PR' ? Why are these two quantities equal? We see that PQ . PR = PA . PB (power property of the blue circle) = PQ'. PR' (power property of the yellow circle). So point P has the same power with respect to both circles. Notice that the argument still holds if P lies at an intersection of the circles, or on the line segment AB. Thus
Now what happens if we have three intersecting circles? 
Three Intersecting Circles
Proof To prove this assertion, suppose that axes BB' and CC' meet in O. Does AA' also pass through O? Since O lies on BB', O has the same power with respect to the green circle and the magenta circle. Since O lies on CC', O has the same power with respect to the blue circle and the magenta circle. Hence, O has the same power with respect to the green circle and the blue circle. Hence O lies on the radical axis AA' of these two circles. Thus the three radical axes meet in the common point O. 
Extensions 1. Let C be a circle of radius r and centre O. Sometimes the power of P with respect to C is defined as OP ^{2} – r ^{2}. How does this compare with the definition we have given? What happens when P lies inside the circle? on the circle? 
Hints and Solutions
1. Let C be a circle of radius r and centre O. Sometimes the power of P with respect to C is defined as OP ^{2} – r ^{2}. How does this compare with the definition we have given? What happens when P lies inside the circle? on the circle? 