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Area of Triangle

Area of triangle Δ = 72631.png = 72625.png = 72618.png
 
Δ = 72612.png = 2R2sin A sin B sin C
 
Δ = 72606.png
 
Δ = r · s

Different circles and centers connected with triangle

Circumcircle and circumcenter (O)
 
The circle passes through the angular point of a ΔABC 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.
     
    73919.png
  • Circumcenter of obtuse-angled triangle lies outside the triangle.
     
    73922.png
  • Circumcenter of right-angled triangle is mid-point of hypotenuse.
     
    73962.png
Distance of circumcenter from sides
 
74009.png
 
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.
 
74069.png
  • Internal bisector AP divides side BC in ratio AB: AC
     
    or 71518.png
     
    BP = ck, CP = bk
    But BP + CP = a ck + bk = a k = 71512.png
    BP = 71505.png71499.png
    Similarly, AQ = 71493.png, CQ = 71487.png
    and AR = 71481.png, BR = 71475.png
  • Area of triangle in terms of r is Δ = rs
  • r = (sa) tan 71469.png = (sb) tan 71463.png = (sc) tan 71457.png
  • r = 4R sin 71451.png sin 71445.png sin 71439.png

Orthocenter

Orthocenter (H) is the point of intersection of altitudes of triangle.
  • Orthocenter (H) of acute-angled triangle lies inside the triangle.
     
    Here A is orthocenter or ΔHBC.
     
    74097.png
  • Orthocenter (H) of obtuse-angled triangle lies outside the triangle.
     
    Here A is orthocenter or ΔHBC.
     
    74130.png
  • Orthocenter (H) of right-angled triangle ABC, right angled at B is B.
     
    74140.png




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