# Thread: Proofs! Externally Tangent Circles

1. ## Proofs! Externally Tangent Circles

2 circles – one of radius 5, the other of radius 8 – intersect at exactly one point and the center of each circle lies outside the other circle. A line is externally tangent to both circles. Find the distance between the two points of tangency. Draw a figure and label it. Justify your answer completely with postulates and theorems. A two-column proof is suggested.

THANKS SO MUCH FOR HELPING ME!!

2. More generally:

Consider two circles with radii $\displaystyle r,R$ then the value of line externally tangent to both circles is given by $\displaystyle 2\sqrt{r\cdot R}$

3. thanks but can you be more specific and include proofs using postulates and theorems?

4. Originally Posted by cbbplanet
thanks but can you be more specific and include proofs using postulates and theorems?
Hello,

I've attached a diagram of the situation. (The light grey circles are necessary to construct the tangent points)

To calculate the distance between the tangent points you are dealing with a right triangle, which I've coloured grey.

The length of the hypotenuse is (R + r)
One leg (coloured violet) has the length (R-r)

Now use Pythagorean theorem:

$\displaystyle d^2 + (R-r)^2 = (R+r)^2$ Expand the brackets:

$\displaystyle d^2 + R^2 - 2rR + r^2 = R^2 + 2rR + r^2$ . Subtract on both sides r² and R²:

$\displaystyle d^2 = 4rR~\implies~d = 2\sqrt{r \cdot R}$

5. I NEED AN EMERGENCY ASSISTANCE!!! the proofs are due tommorrow.

i have figured everything out but can someone tell me how to prove the grey shade triangle is a right triangle? THANKS!!!

6. Originally Posted by cbbplanet
I NEED AN EMERGENCY ASSISTANCE!!! the proofs are due tommorrow.

i have figured everything out but can someone tell me how to prove the grey shade triangle is a right triangle? THANKS!!!
One of its angle is the angles between a tangent and a radius to a circle and hence right.

RonL