# Centroid of Triangle

In Î”

*ABC*, the mid-points of the sides*BC*,*CA,*and*AB*are*D*,*E,*and*F*respectively. The lines*AD*,*BE,*and*CF*are called medians of the triangle*ABC*, the points of concurrency of three medians is called centroid. Generally it is represented by*G*.Also,

*AG*=*AD*,*BG*=*BE*, and*CG*=*CF***Apollonius theorem**

*AB*

^{2}+

*AC*

^{2}= 2(

*AD*

^{2}+

*BD*

^{2})

# Escribed circles of a triangle and their radii

The circle which touches the side

*BC*and two sides*AB*and*AC*produced of a triangle*ABC*is called the escribed circle opposite to the angle*A*. Its radius is denoted by*r*_{1}. Similarly,*r*_{2}and*r*_{3}denote the radii of the escribed circles opposite to the angles*B*and*C*respectively. The centers of the escribed circles are called the ex-centers. The center of the escribed circle opposite to the angle*A*is the point of intersection of the external bisectors of angles*B*and*C*. The internal bisector of angle*A*also passes through the same point. The center is generally denoted by*I*_{1}.In a Î”

*ABC*, we have*r*_{1}=*r*_{1}=*s*tan*r*_{1}= 4*R*sin cos cos ,*r*_{2}= 4*R*cos sin cos ,*r*_{3}= 4*R*cos

**m â€“ n Theorem**

Let

*D*be a point on the side*BC*of a Î”*ABC*such that*BD*:*DC*=*m*:*n*and âˆ*ADC*=*Î¸*, âˆ*BAD*=*Î±*, and âˆ*DAC*=*Î²*. Then- (
*m*+*n*) cot*Î¸*= m cot*Î±*â€“*n*not*Î²* - (
*m*+*n*) cot*Î¸*=*n*cot*B*â€“*m*cot*C*