Two circles with radii a and b touch each other externally . Let c be the radius of a circle which touches these two circles as well as their common tangent to the two circles . Then relate a,b and c through an equation

1/^r =1/^p 1/^q

To relate the radii of the three circles (a, b, and c) through an equation, we can use the concept of "external tangents" and apply the Pythagorean theorem.

Consider the diagram below:
```
* * *
* c * c * c *
* _|_|_|_ *
* * | a | * *
*______*__|_____|__*______*
a b a
```

Here, we have two circle with radii "a" and "b," and a circle with radius "c" that touches both circles externally. The tangent to the two circles is also indicated.

To relate a, b, and c, we can use the following equation:
(a + b)^2 = (a - b)^2 + (2c)^2

Explanation:
1. The total length between the centers of the two smaller circles (a + b) is equal to the difference in radii (a - b) plus twice the radius of the larger circle (2c). This is because the larger circle must touch both the smaller circles externally, which places it at a distance equal to the difference in radii, and also touches the common tangent, which contributes an additional distance equal to twice its radius.

2. By squaring both sides of the equation, we eliminate the square root, which simplifies the equation.

3. Expanding the squares on both sides, we get:
(a^2 + 2ab + b^2) = (a^2 - 2ab + b^2) + 4c^2

4. Simplifying the equation, we get:
4ab = 4c^2

5. Dividing both sides of the equation by 4, we obtain:
ab = c^2

So, the relationship between the radii of the three circles can be represented by the equation ab = c^2.