1. ## touching circles

I am looking at this image, wondering is it obvious that if these two circles touch then there must be a straight line that passes through the centre points?

http://www.cut-the-knot.org/proofs/ford2.gif

Is this based on a circles theorem? Can anyone help please?

2. ## Re: touching circles

Hey rodders.

Try using standard trig [i.e. involving right angles] and basic identities involving circles.

We don't do your work for you.

3. ## Re: touching circles

Thanks. Will give it go. Just wanted a hint.
I am thinking the distance between the two centres is the sum of the radii ? Or thinking there will be one point of intersection between the two circles rather than two ? And do it algebraically? I will think

4. ## Re: touching circles

Yes, when two circles touch at a point, it is easy to show there is a straight line between their centers that passes through that point of intersection.

5. ## Re: touching circles

Is the straight line that passes through the two centres perpendicular to the line that is formed by subtracting the two equations from each other?

6. ## Re: touching circles

What two equations are you subtracting from each other?

7. ## Re: touching circles

Two circle equations.
(x-4)^2 + ( (y-3)^2 =9
and
(x-a)^2 + (y-1)^2 = 1

I am trying to find the value of a so the smaller circle touches the bigger one.

8. ## Re: touching circles

Originally Posted by rodders
Two circle equations.
(x-4)^2 + ( (y-3)^2 =9
and
(x-a)^2 + (y-1)^2 = 1

I am trying to find the value of a so the smaller circle touches the bigger one.
The distance between their centers will be $R + r = 3 + 1 = 4$

large circle has center $(4,3)$, smaller circle has center $(a,1)$

$(a-4)^2 + (1-3)^2 = 4^2$

note there are two possible values for $a$ ...

9. ## Re: touching circles

Thanks.
I managed to work that out too. I am thinking about another circle inbetween the two touching both!
Can't see a simple way of doing this!

http://www.cut-the-knot.org/pythagor...onSangaku1.gif