Given:
- Triangle ABC is a right triangle.
- D is the midpoint of the hypotenuse AB.

To Prove:
Bd = 1/2 * ac

Proof:

1. Introduction:
In this proof, we will show that the length of the line segment Bd is equal to half the product of the lengths of the other two sides of the right triangle ABC, namely ac.

2. Construction:
Let's construct the right triangle ABC with right angle at C and the midpoint D on the hypotenuse AB.

3. Property of Midpoints:
It is a well-known property that the line segment joining the midpoint of the hypotenuse of a right triangle to one of the vertices is half the length of the hypotenuse itself. Thus, we can state that BD = 1/2 * AB.

4. Applying the Property:
Since D is the midpoint of AB, we can replace AB with 2BD in the above equation: 2BD = 1/2 * AB.

5. Simplification:
Simplifying the equation, we get: 2BD = 1/2 * (AC + BC).

6. Substituting the Lengths:
We know that AC is the adjacent side to angle C and BC is the opposite side to angle C. Using the trigonometric ratios, we can express AC and BC in terms of the hypotenuse AB and angle C.

7. Applying Trigonometric Ratios:
From the right triangle ABC, we have:
AC = AB * cos(C)
BC = AB * sin(C)

Substituting these values into equation (6), we get:
2BD = 1/2 * (AB * cos(C) + AB * sin(C))

8. Factoring out AB:
Factoring out AB from the equation, we have:
2BD = 1/2 * AB * (cos(C) + sin(C))

9. Simplifying the Equation:
Canceling out the common factor of 2, we get:
BD = 1/2 * AB * (cos(C) + sin(C))

10. Using Triangle Similarity:
Since triangle ABC is a right triangle, we can use the property of similar triangles. The ratio of the lengths of the sides of similar triangles is equal. Therefore, the ratio of BD to AB is equal to the ratio of AC to BC.

11. Applying the Similarity Property:
Using the similarity property, we get:
BD/AB = AC/BC

12. Substituting the Lengths:
Substituting the lengths from step 6, we have:
BD/AB = (AB * cos(C))/(AB * sin(C))

13. Simplifying the Equation:
Canceling out the common factor of AB, we get:
BD/AB = cos(C)/sin(C)

14. Simplifying the Ratio:
Using the trigonometric identity tan(C) = sin(C)/cos(C), we can rewrite the above ratio as:
BD/AB = 1/tan(C)

15. Recalling the Definition of tan(C):
The tangent of an angle