Prove that opposite sides of a quadrilateral circumscribing a circle subtend supplementary angles at the centre of the circle.

To prove that opposite sides of a quadrilateral circumscribing a circle subtend supplementary angles at the center of the circle, let's break it down step by step.

Step 1: Understanding the Quadrilateral and the Circle
Let ABCD be a cyclic quadrilateral (a quadrilateral that circumscribes a circle). This means that there exists a circle which touches all four sides of the quadrilateral. The circle is called the incircle of the quadrilateral. The point where the circle touches the sides of the quadrilateral is known as the point of tangency.

Step 2: Properties of Tangents to a Circle
The tangent to a circle at any point is perpendicular to the radius drawn to the point of tangency.
In this case, the quadrilateral ABCD has a circle inscribed in it, and the points where the sides touch the circle are denoted as P, Q, R, and S, where:
P is the point of tangency of side AB,
Q is the point of tangency of side BC,
R is the point of tangency of side CD, and
S is the point of tangency of side DA.
Step 3: Angles at the Center of the Circle
Let the center of the circle be O. We need to prove that opposite sides of the quadrilateral subtend supplementary angles at the center O. Specifically, we need to show that:

The angle between the lines OA and OC (which connects the center O to the endpoints of opposite sides AB and CD) is supplementary to the angle between OB and OD (which connects the center O to the endpoints of opposite sides BC and DA).
Step 4: Angle Relationships
The angle between any two radii of a circle is equal to the central angle subtended by the arc between those two radii.
In quadrilateral ABCD, the opposite sides AB and CD are tangent to the circle at points P and R, respectively. Similarly, BC and DA are tangent at points Q and S, respectively.
Let’s focus on the angle between opposite sides:

The central angle subtended by side AB (let’s call it ∠AOB) corresponds to the arc that joins points P and Q on the circle.
Similarly, the central angle subtended by side CD (let’s call it ∠COD) corresponds to the arc that joins points R and S on the circle.
Since the opposite sides AB and CD are tangents, the arc subtended by AB plus the arc subtended by CD will cover the entire circumference of the circle. Therefore, the sum of the central angles ∠AOB and ∠COD is 180°.

This relationship holds similarly for the angles formed by sides BC and DA. The central angle subtended by BC plus the central angle subtended by DA is also 180°.

Step 5: Conclusion
Thus, opposite sides of the quadrilateral subtend supplementary angles at the center of the circle. In other words, the angle between the lines connecting the center of the circle to opposite vertices of the quadrilateral is supplementary.

This proves the given statement.