I recently came across an intriguing geometry problem from the Bangladesh Math Olympiad $2018$, and I'm seeking some assistance in solving it. This problem involves a circle with a unique configuration of tangents and an exciting twist. It's an excellent challenge for geometry enthusiasts, and I'd appreciate your insights and expertise in finding the solution.
Problem Description
We have a circle with a diameter $AB$, and two tangents, $AD$ and $BC$, drawn to the circle in such a way that they intersect at a point on the circle itself. Furthermore, it's provided that $AD = a$ and $BC = b$, with the condition $a≠b$. The ultimate goal is to determine the diameter AB of the circle based on these given conditions.
My Approach: I consider the intersecting point $O$. now as the $△ADO$ is a right-angle triangle. If somehow I could express the length $AO$ with $a$ & $BO$ with $b$ I easily express the diameter $AB$. But here I can't figure out how to express those $AO$ & $BO$ with those tangents. To do so I also need $OD$ & $OC$ .
If anyone can give any better opinion or better solution please lend me your valuable opinion and insights.
Problem Source:
Bangladesh Math Olympiad $(2018)$ ; Round: National ; Category: Higher Secondary
I look forward to your valuable input and solutions for this intriguing geometry problem. Thank you for your assistance!
$\triangle OAD\sim \triangle OBA$ (AAA similarity)
$\frac{OA}{OB}=\frac{AD}{AB}=\frac{a}{AB}$ (sides are in a proportion)
$\triangle OBC\sim \triangle OAB$ (AAA similarity)
$\frac{OA}{OB}=\frac{AB}{BC}=\frac{AB}{b}$ (sides are in a proportion)
then
$\frac{a}{AB}=\frac{AB}{b}$
$AB=\sqrt{ab}$