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
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
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’.