What you have to do, depends on the definition you use.
The diameter (length) of a closed bounded shape like a circle, is the
maximum of the
distances between any two points on the shape.
If we take that as the definition of the diameter of a circle, then we can try to prove that the area is bisected by a diameter line.
Consider a
diameter line AB, to be a line between 2 points A, B on the circle, which has that maximum, diameter length,
d. We might prove that AB goes through the centre of the circle.
Suppose it does not. Then we can draw a triangle ABC:
with BC≠0.
It is easily proved that the angle ABC is 90º
Now [MATH]AC^2=AB^2+BC^2[/MATH][MATH]\implies[/MATH] [MATH]AC>AB[/MATH] (since[MATH] BC≠0[/MATH])
Since A, C are points on the circle and AB is a diameter line of length
d, this contradicts the fact that
d is the maximum distance between 2 points on the circle.
Therefore, any diameter line goes through the centre.
Now you can use e.g. Dr.Peterson's symmetry argument, to show that the area of the circle is bisected by the diameter line.