circles with collinear points

[kaknutson2]

Take three circles of different radii, none of which is contained inside another. Draw the two common external tangents to each pair of circles, and let P, Q and R be the points where each pair meets. Show that P, Q, and R are collinear.

I really have no idea where to start, so any help would be greatly appreciated! Thank you!

 

[tkhunny]

Draw three circles. They can intersect, but none can be entirely contained in any other. (Note #1) Make sure they are not the same size. (Note #2)

IF you drew them so that they do NOT intersect, you will have too kinds of common tangents. 1) External. Think of a bicycle chain. 2) Internal. More of a figure-eight sort of thing. If your circles intersect at all, you will not have any common internal tangent.

Draw the three pairs of EXTERNAL common tangents. Each pair of circles will have two.

For each pair of circles and their corresponding external common tangents, extend the common external tangents in the direction that will result in their intersection. This should define your three points.

Now what?

Note #1 - If one is contained in another, there will be no common tangent. Anything tangent to the one contained must necessarily intersect the one doing the containing. This assumes the circles share no boundary. That's a slightly different problem.

Note #2 - If they are the same size, the tangents will be parallel. Where does that mean they will intersect?

 

[kaknutson2]

I've drawn the three circles and can see that the three points should be collinear (and if they would be parallel they intersect at a point infinity). Now I just need to find why they are collinear!