Math Olypiad Question!!!!?

Question

"Two circles G1 and G2 are contained inside circle G, and are tangent to G at the distinct point M and N, respectively. G1 passes through the center of G2. The line passing through the two point of intersection G1 and G2 meets G at A and B. The lines MA and MB meet G1 at C and D, respectively.

Prove that CD is tangent to G2."

I've been working on this for a little more than an hour and I couldn't even get started. I really wanted to do this on my own, but I couldn't; so please just give me the theorems or laws needed. Thanks!

Thanks! I drew it, but it's just too small to understand. And the lines suck. Well I did get started, just not starting to do the problem.

Currently I found out what I'm suppose to prove, but I've been stuck trying find slope of AB.

Wait a sec, how do you know line AB is perpendicular to the radius of G2?

Okay, wrong question xD! How do you know AB is parallel to CD?

I'm sorry, my English sucks so bad I think I'm reading the SAT's. Zo Maar, can you explain to me what homothetical transofrmation is without using these weird words... I got the power of the point theorem, but not so much the other parts..

And are those lines absolute values? ex. |AB|, |CD|, etc..

Sorry, but I'm only into math. Anything else I'm near a 5th grader.

Thanks I understand now. The real problem is now to solve the problem... I'm going to try to do this one even it it takes me a year.

Answer 1

EDIT:

Homothety is a stretching (or re-scaling). Imaging, you have a piece of rubber. The point M is fixed. Then you stretch the rubber in such a way that every point A moves along the line AM, connecting it to point M. If the distance between points A and M is |AM| (I use |...| for distances), then point A is transformed into point C, such as the distance between C and M is equal to k|AM|. k is the strectching factor (or ratio), which is the same to all the points.


What you need to know for solving your problem is that by applying this stretching to tangent circles you can transform the circles one into the other. Then you start stretching circle G1, keeping the point M fixed (or vice versa you can shrink G to G1). Soon or later G1 will coincide with G. You also know that all the points move along straight lines, connecting them with the point M, so that C is transformed into A, D is transformed into B etc.



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Doctor D is almost right. This is about transformations. Only you should apply homothetical transformations rather than inversions.

A homothety with the center M and ratio k maps each point A to the point C lying on line MA, such as |MC|/|MA|=k. Homothety is a similarity transformation. If two circles are tangent at point M, then there exists a homothety mapping one circle into the other.

You also need the power of a point theorem. If we have a circle G1 and point A, then for any line passing through A and intersecting G1 at points C and M, the product |AC|*|AM| is the same.

Consider homotheties with centers M and N, which map G1 and G2 into the big circle G. You can observe some properties of lines, e.g. that AB is parallel to CD (because |MC|/|MA|=|MD|/|MB|=k). You can also show that the line tangent to circle G1 at point C is also tangent to G2 (let this line touches G2 at point F).

Then, if you prove that the center of G2 lies on bissect of the angle FCD, your problem will be solved (I prefer this way).

Alternatevely, you can use the information you accumulated by considering homotheties to calculate the distance from the center of G2 to the line CD. It should be equal to the radius of G2.

If there are still problems, I can give you complete solution.

Answer 2

this is what it looks like!!!:

http://i167.photobucket.com/albums/u141/SitrONes7/geometryproblem.jpg

[oops! sorry the circles aren't labeld right! big circle is G, middle circle is G1, small circle is G2]


interesting huh? Wish I had auto cad so i could draw this easier. I'm gonna think about this one later. Maybe I could find an approach.

My personal advice to you is: Don't challange yourself too hard! It is not healthy. Progress as you gain more experience with geometry. My father tells me, "don't read something you know you're not gonna understand, it'll only make you feel stupid!" I think he is right.

Answer 3

Check out this question asked by Dr. Dumbfellow and answered by Zo Maar. It may be similar to yours. See if it helps.

http://answers.yahoo.com/question/index;_ylt=Ar2uUvYHthXDN2zwGnBUtUDty6IX;_ylv=3?qid=20071010081514AAm63KN