The diameter is the longest cord in the circle
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pka
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If [imath]\theta[/imath] is the measure of a central angle is a circle of radius [imath]\mathit{R}[/imath]
then the length of the subtended chord is [imath]{\bf c}=2\mathit{R}\cdot\sin\left(\dfrac{\theta}{2}\right)[/imath]
The length of a diameter is [imath]D=2\mathit{R}[/imath]
The size of a cental angle determined by a diameter is [imath]\pi\text{ radians}[/imath]
[imath][/imath][imath][/imath][imath][/imath]
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blamocur
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You can look at the triangles formed by a cord and two radii, then use the triangle inequality.
insanecarrots
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There is a diameter, and here is a lovely little evidence I devised decades ago.
Make a circle O with any chord AB on it. Draw a line segment through the centre of the chord from one endpoint, say A. That is, draw a circle. View the diagram. Draw a radius from centre O to centre B.
According to the triangle inequality, AB AO + OB = r + r =2r = d. As a result, every chord that is NOT a diameter is less than a diameter. As a result, the largest chord is a diameter.
