## 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**.

**Solution**

**Discussion**:

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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