Solution Insights: Problem on Inequalities for 27–01–2022
Published in
Jan 27, 2022
This problem has a simple solution that requires the knowledge of two elementary facts — the geometric property of intersecting chords of a circle, and the relation between the Arithmetic Mean and the Geometric Mean of a set of numbers.
The first fact states that if AB and CD are two chords in a circle and intersect each other at P, then (AP)(PB) = (CP)(PD). From this, we get the first essential fact to solve this problem:
The second fact states that the Arithmetic Mean of a set of numbers is always either greater than or equal to their Geometric Mean. Thus, we get the second essential fact to solve the problem:
This directly leads us to the relation which we had wanted to prove,
QED.
