In the figure, radii divide the opposite angles into parts. Using the isosceles triangles formed by equal radii, we get p + v = 90° and q + u = 90°. Hence, opposite angles sum to 180°.
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The diameter passes through the centre and is the longest chord of a circle. Any other chord lies away from the centre and is therefore shorter than the diameter. Hence, no chord is longer.
∠MOP is the angle subtended by arc MP at the centre, while ∠MNP is subtended by the same arc at the circumference. Therefore, the central angle is twice the angle at the circumference.
Each side of a regular hexagon subtends 60° at the centre. The triangle formed is equilateral, so each side is r. The distance from the centre to each side is (√3/2)r.
All chords have the same length, so their midpoints are at the same distance from the centre of the circle. Therefore, the midpoints of all such chords form a circle concentric with the given circle.