Home > Circles, Triangles > Triangle – Circle problem

Triangle – Circle problem

Given triangle ABC. \angle C = 90^\circ . CD is height of triangle ( point D is on hypotenuse AB ). The radiuses of incircles of triangles ACD and ABD are r_1 \: and \: r_2 . Find the radius of incircle of triangle ABC .

Solution :

Let O_1 be center of incircle of triangle ACD with radius r_1 and O_2 be center of incircle of triangle BCD with radius r_2 . Let the center of incircle of triangle ABC O_3 with radius R. ( we have to find R ) And let AC=b, CB=a, AB=c .

The center of incircle is the intersection point of bisectors of triangle.

\triangle ACO_1 \sim \triangle BCO_2 \sim \triangle ABO_3

\frac{ r_1 }{R} = \frac{b}{c} \: and \: \frac{ r_2 }{R} = \frac{a}{c}

\frac{ r^2_1 + r^2_2 }{ R^2 } = 1 \Rightarrow R= \sqrt{ r^2_1 + r^2_2 }

 

CD = R + r_1 + r_2

 

Advertisements

Leave a Reply

Please log in using one of these methods to post your comment:

WordPress.com Logo

You are commenting using your WordPress.com account. Log Out / Change )

Twitter picture

You are commenting using your Twitter account. Log Out / Change )

Facebook photo

You are commenting using your Facebook account. Log Out / Change )

Google+ photo

You are commenting using your Google+ account. Log Out / Change )

Connecting to %s

%d bloggers like this: