A Mean Problem

Consider two segments of lengths a and b. Construct two new segments, one half the difference in these two lengths and one half the sum. Use half the difference as the radius of a circle at P. Construct PM as the half sum. Clearly PM represents the arithmetic mean of a and b.

Construct a tangent from the circle from M to obtain the segment MN.

Construct a perpendicular from P to the circle at R and construct RM.

Construct perpendicular from N to PM. Label the intersection S.

Find MN in terms of a and b.

Find MR in terms of a and b.

Find MS in terms of a and b.



From the construction it is clear that when a > b,

MS < MN < MP < MR

and they would be equal if a = b. Interpret.

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