# Area of Triangle

Area of triangle Î” = = =

Î” = = 2

*R*^{2}sin*A*sin*B*sin*C*Î” =

Î” =

*r*Â·*s*# Different circles and centers connected with triangle

*Circumcircle and circumcenter (O)*

The circle passes through the angular point of a Î”

*ABC*is called its circumcircle. The center of this circle is the point of intersection of perpendicular bisectors of the sides and called the circumcenter. Its radius is denoted by*R*.- Circumcenter of acute angled triangle lies inside the triangle.
- Circumcenter of obtuse-angled triangle lies outside the triangle.
- Circumcenter of right-angled triangle is mid-point of hypotenuse.

*Distance of circumcenter from sides**OD*

*=*

*R*cos

*A*,

*OE*=

*R*cos

*B*, and

*OF*=

*R*cos

*C*

# Incircle and incenter (I)

Point of intersection of internal bisectors of triangle is incenter of triangle.

Also it is center of the circle touching all the three sides internally.

Incenter always lies inside the triangle.

- Internal bisector
*AP*divides side*BC*in ratio*AB*:*AC**BP*=*ck*,*CP*=*bk*But*BP*+*CP*=*a*â‡’*ck*+*bk*=*a*â‡’*k*=â‡’*BP*=Similarly,*AQ*= ,*CQ*=and*AR*= ,*BR*= - Area of triangle in terms of
*r*is Î” =*rs* -
*r*= (*s*â€“*a*) tan = (*s*â€“*b*) tan = (*s*â€“*c*) tan -
*r*= 4*R*sin sin sin

# Orthocenter

Orthocenter (

*H*) is the point of intersection of altitudes of triangle.- Orthocenter (
*H*) of acute-angled triangle lies inside the triangle.*A*is orthocenter or Î”*HBC*. - Orthocenter (
*H*) of obtuse-angled triangle lies outside the triangle.*A*is orthocenter or Î”*HBC.* - Orthocenter (
*H*) of right-angled triangle*ABC*, right angled at*B*is*B*.