This is problem number 5 from Indian Statistical Institute, ISI BStat 2005 based on Geometric inequality.

**Consider an acute angled triangle PQR such that C,I and O are the circumcentre, incentre and orthocentre respectively. Suppose and , measured in degrees, are and respectively. Show that **

Discussion:

Use this diagram. Angle chasing should suffice. Assume the that the measure of

**Student:** since it is the angle at the center (twice the angle at circumference). Again

Finally

Hence

But here is a problem. If I maximize P (by replacing it 90), it adversely affects 180 – P as Pis negative there. Somehow we must balance the value of P.

**Teacher: **Right. If you replace P by 90, 1/2P will remain greater than 1/180, but 1/90 will be less than 1/180-P.

Here we need to use Arithmetic Mean > Harmonic Mean inequality which state for positive numbers a, b, c we have

**Student:** Ok. I think I can do it from here.

Since it is an acute angled triangle,

Hence replacing P by 90, we get

**Teacher:** Just one point of caution; all numbers are in degrees. Also mention that ‘increasing the denominator decreases a number’.

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