Math Help - Circles and lines! Really interesting problem!!

1. Circles and lines! Really interesting problem!!

Consider 3 circles with different radii. For any 2 of them one can draw 2 common tangent lines, convergent to points A, B and C.

Prove: A, B and C are on a straight line.

(and what makes it interesting? you ask. Well, I just want to see how many DIFFERENT proofs I can get! I have one in mind)

2. Re: Circles and lines! Really interesting problem!!

Well... nobody responds... let me give a proof I like:

Proof: Suppose the radii are ${r}_{1},\ {r}_{2},\ {r}_{3}$. Now instead of circles, imagine three BALLS with radii ${r}_{1},\ {r}_{2},\ {r}_{3}$. Place them in such a way that 1) centers of the balls are on a same plane (call this plane X) and 2) any ball touches the other two. Wrap 2 balls with a cone as in the attachment. For the three balls then we can wrap 3 cones. Ajust the vertex of the cones so that all of them falls on plane X. Notice that now the balls and cones are symmetric about plane X. Let plane Y be the plane that touches the 3 balls on one side of X and plane Z touches the 3 balls on the other side of X. Then the intersection $\l$ of Y and Z falls on X, too, because of the symmetry. Clearly, A, B, C $\in$ $\l$. Q.E.D.

---------------------------------------------------------------------

The proof can be made cleaner and more rigorous. But we see the point... By escalating the problem to a higher dimension we actually make it EASIER to solve. I have this feeling not just for this particular problem: by engaging ourselves more than a problem seems to require is often a better way to work it out... paradoxically lovely idea

3. Re: Circles and lines! Really interesting problem!!

Hi thank you for sharing this puzzling math problem
I couldn't help with this though... Would love to hear other thoughts on this. Thanks!

----------------------------------------------------------------
Dog Crates, Pet Crates. 30% OFF & FREE Shipping on Pet Crates and Dog Crates