# Circle Geometry

## Homework Statement

There is a semicircle with radius 1. Two circles are inscribe in it with centres C1 and C2 and radius r1 and r2 respectively. Find the maximum possible value of r1+r2
Here is the picture, I have drawn.
http://img143.imageshack.us/img143/8392/circlesdn3.png [Broken]

## The Attempt at a Solution

I have constructed the tangents as shown in the figure. I have shown the right angles. And I also deduce that:
AF=DF=CF
From this I see that r1 is necessarily equal to r2. Isnt it?? But I am not sure if I have drawn DF right. It might not be a tangent to both circles in some cases??

Help me!

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Dick
Homework Helper
I think I'm getting this. Pick the center of one of your circles. Can you show it lies on an ellipse? It's center is equidistant between the semicircle and the x-axis (the diameter of your semicircle). Pick C=(x,y) and find the conditions that imposes. Now concentrate on the quadrilateral formed by the centers of the two circles and the vertical lines dropping to the x-axis. That can give a condition relating the two circle radii. This is not an easy problem, as near as I can tell. It takes some work to put it together. Can you start this out? I'm only going to give hints.

If the semi circle lies on the x axis, and if you have two circles with centers c1 and c2 with radii r1 and r2 (r1>r2) and you join the line joining the centers of the circles and extend it so that it intersects the x axis, and it makes an angle $$\theta$$ with the x axis, you can come up with the following relation,

$$(r_1+r_2)sin\theta=r_1-r_2$$

where do you go from here?

Avodyne