Explanation of ratio between radii of circles

Question

Given two circles with centers and radii of $C_{0},C_{1}$ and $R_{0},R_{1}$ respectively, as shown in the following diagram:

http://i.imgur.com/iBOOhaM.png

The red point $I$ is the intersection point of the internal tangent lines between the two circles.

Let's define the lengths $N_{0}$ and $N_{1}$ as:

$N_{0} =|C_{0} - I|$, distance between $C_{0}$ and $I$

$N_{1} =|C_{1} - I|$, distance between $C_{1}$ and $I$

My question: How is the following equality derived?

$$\frac{R_{0}}{R_{1}} = \frac{N_{0}}{N_{1}}=m$$

---

As a side note, Let's call $D$ the distance between $|C_{0} - C_{1}|$, which is also:

$$D=R_{0}+|K_{0}-I|+R_{1}+|K_{1}-I|=N_{0}+N_{1}$$

this then leads to the following:

$$N_{0} = \frac{mD}{m+1}$$ $$N_{1} = \frac{D}{m+1}$$

Which then leads to the definition of the point $I$ as:

$$I=N_{0}|C_{1} - C_{0}|=N_{1}|C_{0} - C_{1}|$$

Answer 1

Draw one of the internal tangents through I and tangent to both circles. Then from the center of each draw the radius to the points of tangency for each circle. That will give you two similar right triangles since they each share two congruent angles. In similar triangles the ratios of corresponding sides are equal thus

$$\frac{R_{0}}{N_{0}} = \frac{R_{1}}{N_{1}}$$

from which it follows that

$$ \frac{R_{0}}{R_{1}} = \frac{N_{0}}{N_{1}} $$

Answer 2

Let $T_0$ and $T_1$ be the respective points of tangency of one of the tangent lines. Then $\triangle{C_0T_0I}$ is similar to $\triangle{C_1T_1I}$.