Problem:
A, b, and c are the sides of a right triangle, where c is the hypotenuse. A circle, of radius r, touches the sides of the triangle. Prove that r = (ab - c)/2.

Solution:

We are given a right triangle with sides A, B, and C, where C is the hypotenuse. Let's label the vertices of the triangle as follows:

- The right angle vertex as O.
- The vertex on side A as P.
- The vertex on side B as Q.

Step 1: Drawing the Triangle:
Draw a right triangle ABC, with C as the hypotenuse. Next, draw the circle inscribed in the triangle, touching sides A, B, and C. Let's label the center of the circle as O and the points where the circle touches the sides as P and Q.

Step 2: Identifying Key Points:
Let's identify some key points:

- The radius of the circle, which is represented by r.
- The length of side A, which is represented by a.
- The length of side B, which is represented by b.
- The length of side C, which is represented by c.

Step 3: Identifying Tangents:
Since the circle is inscribed in the triangle, the tangent to the circle at points P and Q will be perpendicular to sides A and B, respectively. Let's draw these tangents.

Step 4: Finding Lengths of Tangents:
The length of the tangent from the vertex P to the circle is equal to the length of side A, which is a. Similarly, the length of the tangent from the vertex Q to the circle is equal to the length of side B, which is b.

Step 5: Applying Tangent-Secant Theorem:
By applying the Tangent-Secant Theorem, we can find the length of the secant. According to the theorem, the product of the lengths of a secant segment and its external segment is equal to the square of the length of the tangent segment.

Applying this theorem to our triangle, we have:

(a + b) * (a - b) = c^2

Expanding the equation, we get:

a^2 - b^2 = c^2

Rearranging the equation, we have:

a^2 = b^2 + c^2

Step 6: Applying the Pythagorean Theorem:
Since we have a right triangle, we can apply the Pythagorean Theorem, which states that the square of the hypotenuse is equal to the sum of the squares of the other two sides.

Applying the Pythagorean Theorem to our triangle, we have:

c^2 = a^2 + b^2

From Step 5, we know that a^2 = b^2 + c^2. Substituting this into the Pythagorean Theorem equation, we get:

c^2 = (b^2 + c^2) + b^2
c^2 = 2b^2 + c^2
0 = 2b^2

Simplifying the equation, we have:

b^2 = 0

Therefore, the only possible solution