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Maths proof

How do you prove that for any circle in a coordinate grid, the line through the circle and the origin goes through the closest point from the edge of the circle to the origin?

Reply 1

Kevin De Bruyne

Original post by muttonrolls

How do you prove that for any circle in a coordinate grid, the line through the circle and the origin goes through the closest point from the edge of the circle to the origin?

To me, 'the line through the circle and the origin' sounds a bit unclear.

I have asked for your thread to be moved to Maths

Reply 2

atsruser

Original post by muttonrolls

How do you prove that for any circle in a coordinate grid, the line through the circle and the origin goes through the closest point from the edge of the circle to the origin?

I guess you mean "the line through the origin and the centre of the circle".

If so, then construct that line, and let it meet the circle at P. Then construct a line from the origin to any other point on the circle, and let it meet the circle at Q. Look at the triangle OPQ. It is clear that the largest angle in OPQ is

O\hat{P}Q

since it is greater than 90 degrees (since the tangent at P already makes 90 degrees and it's bigger than that.). Hence it faces the longest side of the triangle OQ. Thus OP < OQ.