Proof that an isosceles trapezium is always cyclic and its diagonals are equal:

An isosceles trapezium is a quadrilateral with two parallel sides and two non-parallel sides, where the non-parallel sides are equal in length. In this proof, we will show that an isosceles trapezium is always cyclic, meaning that all four of its vertices lie on a common circle. Additionally, we will demonstrate that the diagonals of an isosceles trapezium are equal in length.

Proof:

1. Construct the trapezium:
Start by constructing an isosceles trapezium ABCD, where AB and CD are parallel sides, and AD = BC.

2. Extend the sides:
Extend the non-parallel sides AD and BC to meet at a point E.

3. Observe the triangles:
Notice that triangles ABE and CDE are congruent by the side-side-side (SSS) congruence criterion, as AB = CD (given), AD = BC (definition of an isosceles trapezium), and AE = CE (both are extensions of the parallel sides).

4. Prove the angles:
Since triangles ABE and CDE are congruent, their corresponding angles are also equal. Therefore, angle AEB = angle CED.

5. Prove the cyclic property:
Using the property that opposite angles in a cyclic quadrilateral are supplementary, we can conclude that angle AED + angle BEC = 180 degrees.

6. Prove the diagonals are equal:
Since opposite angles in a cyclic quadrilateral are supplementary, we have angle AED + angle BEC = 180 degrees. This implies that angle AED = angle BEC.

7. Observe the triangles again:
Notice that triangles ADE and BCE are congruent by the angle-side-angle (ASA) congruence criterion. Angle AED = angle BEC (proved in the previous step), AD = BC (definition of an isosceles trapezium), and AE = CE (both are extensions of the parallel sides).

8. Prove the length of diagonals:
Since triangles ADE and BCE are congruent, their corresponding sides are also equal. Therefore, DE = BE.

9. Conclude:
We have shown that an isosceles trapezium ABCD is cyclic because its opposite angles are supplementary. Additionally, the diagonals AD and BC are equal in length as proved by the congruence of triangles ADE and BCE.

Therefore, an isosceles trapezium is always cyclic, and its diagonals are equal in length.