Prove that the diameter is the greatest chord of a circle

Proposition:

Let O be the centre of the circle ABCD. Let AB be the diameter and CD be a chord other than diameter of the circle. It is required to prove that AB > CD.

Construction:

Join O,C and O,D.

Proof:

OA = OB = OC = OD (radius of the same circle)
Now, in ΔOCD,
OC + OD > CD
or, OA + OB > CD
Therefore, AB > CD. (Proved)

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