Explanation:

In a trapezium ABCD where AB is parallel to CD, we need to find the relationship between AC^2, BD^2, BC^2, AD^2, and AB * CD.

Properties of a Trapezium:
1. In a trapezium, the opposite sides are parallel.
2. The diagonals of a trapezium divide it into four triangles.

Using the Properties of a Trapezium:
1. Let's consider the diagonals AC and BD.
2. The diagonal AC divides the trapezium into two triangles, namely △ABC and △ACD.
3. Similarly, the diagonal BD divides the trapezium into two triangles, namely △ABD and △BCD.

Applying the Pythagorean Theorem:
1. We can apply the Pythagorean theorem to each of the four triangles formed by the diagonals.
2. For △ABC: AC^2 = AB^2 + BC^2
3. For △ACD: AC^2 = AD^2 + CD^2
4. For △ABD: BD^2 = AB^2 + AD^2
5. For △BCD: BD^2 = BC^2 + CD^2

Combining the Equations:
1. From equations (2) and (3) above, we can equate AC^2 in both equations:
AD^2 + CD^2 = AB^2 + BC^2

2. Rearranging the equation, we get:
AD^2 + CD^2 - AB^2 - BC^2 = 0

3. Adding AB * CD to both sides of the equation:
AD^2 + CD^2 - AB^2 - BC^2 + AB * CD = AB * CD

4. Simplifying the equation:
AD^2 + AB * CD + CD^2 - AB^2 - BC^2 = AB * CD

5. Rearranging the terms, we get:
AD^2 + AB * CD - AB^2 - BC^2 + CD^2 = AB * CD

6. Finally, we can rewrite the equation as:
BC^2 + AD^2 = AB * CD + AB^2 + BC^2 + AD^2 + CD^2

7. Simplifying further:
BC^2 + AD^2 = AB^2 + CD^2 + 2 * AB * CD

Therefore, in a trapezium ABCD where AB is parallel to CD, AC^2 + BD^2 = BC^2 + AD^2 + 2 * AB * CD.