Let the lengths of the three sides of the triangle ABC be $\mathrm { AB } = 6 , \mathrm { BC } = 8$ and $\mathrm { CA } = 4$. Let $\mathrm { O } ^ { \prime }$ be the center of the circle which passes through the two points B and C and is tangent to the straight line AB. Let O be the center of the circle circumscribed about triangle ABC. We are to find the length of the line segment $\mathrm { OO } ^ { \prime }$. (1) First, we have $\cos \angle \mathrm { ABC } = \frac { \mathbf { A } } { \mathbf { B } }$ and $\sin \angle \mathrm { ABC } = \frac { \sqrt { \mathbf { C D } } } { \mathbf { E } }$. (2) The radius of the circle circumscribed about triangle ABC is $\frac { \mathbf { F G } \sqrt { \mathbf { H I } } } { \mathbf { J K } }$. (3) When the intersection point of the straight line $\mathrm { OO } ^ { \prime }$ and the side BC is denoted by D, we have $$\mathrm { OD } = \frac { \mathbf{N} \sqrt { \mathbf { L M } } } { \mathbf { O P } } \text { and } \mathrm { O } ^ { \prime } \mathrm { D } = \frac { \mathbf { Q R } \sqrt { \mathbf { S T } } } { \mathbf { U V } } .$$ Thus we have $\mathrm { OO } ^ { \prime } = \frac { \mathbf { W } \sqrt { \mathbf { X Y } } } { \mathbf { Z } }$.
Let the lengths of the three sides of the triangle ABC be $\mathrm { AB } = 6 , \mathrm { BC } = 8$ and $\mathrm { CA } = 4$. Let $\mathrm { O } ^ { \prime }$ be the center of the circle which passes through the two points B and C and is tangent to the straight line AB. Let O be the center of the circle circumscribed about triangle ABC. We are to find the length of the line segment $\mathrm { OO } ^ { \prime }$.
(1) First, we have $\cos \angle \mathrm { ABC } = \frac { \mathbf { A } } { \mathbf { B } }$ and $\sin \angle \mathrm { ABC } = \frac { \sqrt { \mathbf { C D } } } { \mathbf { E } }$.\\
(2) The radius of the circle circumscribed about triangle ABC is $\frac { \mathbf { F G } \sqrt { \mathbf { H I } } } { \mathbf { J K } }$.\\
(3) When the intersection point of the straight line $\mathrm { OO } ^ { \prime }$ and the side BC is denoted by D, we have
$$\mathrm { OD } = \frac { \mathbf{N} \sqrt { \mathbf { L M } } } { \mathbf { O P } } \text { and } \mathrm { O } ^ { \prime } \mathrm { D } = \frac { \mathbf { Q R } \sqrt { \mathbf { S T } } } { \mathbf { U V } } .$$
Thus we have $\mathrm { OO } ^ { \prime } = \frac { \mathbf { W } \sqrt { \mathbf { X Y } } } { \mathbf { Z } }$.