## A smart geo

For discussing Olympiad level Geometry Problems
Posts: 12
Joined: Sat Jan 23, 2016 7:55 pm

### A smart geo

Two circles touch internally and the radius of the larger circle is 8 units. Centre of
the larger circle lies on the smaller circle. Diameter of the larger circle that passes
through the touching point meets the larger circle at point A. Tangent drawn from
A to the smaller circle touches that at B. Length of AB is of the form a/b*√2 where a
and b are co-prime. Find a - b.

Absur Khan Siam
Posts: 53
Joined: Tue Dec 08, 2015 4:25 pm
Location: Bashaboo , Dhaka

### Re: A smart geo

The problem statement is incomplete. The radius/
diameter of the smaller circle should be given.$AB$
can be determined in respect of the smaller circle's
Let the radii of the bigger and smaller circle are respectively $O_1$ and $O_2$ and the touching
point of the two circle is $T$.Using the secant-tangent
theorem on the smaller circle,we can write,
$AB^2 = AM \times AT = AT(AT - MT) = 16(16 - MT)$
$\therefore AB = 4\sqrt{16 - MT}$
"(To Ptolemy I) There is no 'royal road' to geometry." - Euclid