Triangle and Tangent Circle  Problem With Solution
A problem, on a triangle tangent at two points to a circle, is presented along with detailed solution.
Problem : In the figure below, triangle ABC is tangent to the circle of center O at two points. The lengths of AM and BC are equal to 6 and 18 cm respectively. Find the radius of the circle.
Solution to Problem :

Let B and N be the two points of tangency of the circle (see figure below). We then have three right triangles. We first use Pythagora's theorem to triangle AON. Let r be the length of the radius of the circle. Hence
AN^{ 2} + r^{ 2} = (r + 6)^{ 2}

Solve the above fo AN
AN = SQUARE ROOT [ 36 + 12 r]

Also triangles ONC and OBC are right triangles such that ON = OB and therefore congruent. Hence NC = BC = 18 cm. Use Pythagora's theorem to triangle ABC
(6 + 2r)^{ 2} + 18^{ 2} = (18 + AN)^{ 2}

Expand the above equation and group like terms
4 r ^{ 2} + 12 r = 36 AN

Substitute AN by SQUARE ROOT [ 36 + 12 r] and divide all terms by 4
r ^{ 2} + 3 r = 9 SQUARE ROOT [ 36 + 12 r]

Square both sides
( r ^{ 2} + 3 r )^{ 2} = 9^{ 2} [ 36 + 12 r]

Expand and group
r ^{ 4} + 6 r ^{ 3} + 9 r ^{ 2}  972 r  2916 = 0

The above equation has two real solutions. Only one of them is positive and is equal to 9 cm.
More references to geometry problems.
Geometry Tutorials, Problems and Interactive Applets.

