Triangle Circle

The Incenter of a triangle

One of several centers the triangle can have, the incenter is the point where the angle bisectors intersect. The incenter is also the center of the triangle's incircle - the largest circle that will fit inside the triangle.

The triangle's incenter is always inside the triangle.

The Circumcenter of a triangle

The circumcenter is the point where the perpendicular bisectors of a triangle intersect. The circumcenter is also the center of the triangle's circumcircle - the circle that passes through all three of the triangle's vertices.

Special case - right triangles

In the special case of a right triangle, the circumcenter (C in the figure at right) lies exactly at the midpoint of the hypotenuse (longest side). See also Circumcircle of a triangle.


Centroid of a Triangle

The point where the three medians of the triangle intersect. The 'center of gravity' of the triangle 
One of a triangle's points of concurrency.

The centroid of a triangle is the point through which all the mass of a triangular plate seems to act. Also known as its 'center of gravity' , 'center of mass' , or barycenter.

Centroid facts

• The centroid is always inside the triangle
• Each median divides the triangle into two smaller triangles of equal area.
• The centroid is exactly two-thirds the way along each median.
  Put another way, the centroid divides each median into two segments whose lengths are in the ratio 2:1, with the longest one nearest the vertex.

Orthocenter of a Triangle

The point where the three altitudes of a triangle intersect. One of a triangle's points of concurrency.

The altitude of a triangle (in the sense it used here) is a line which passes through a vertex of the triangle and is perpendicular to the opposite side. There are therefore three altitudes possible, one from each vertex. See Altitude definition.

It turns out that all three altitudes always intersect at the same point - the so-called orthocenter of the triangle.

The orthocenter is not always inside the triangle. If the triangle is obtuse, it will be outside.

The Euler line - an interesting fact

It turns out that the orthocenter, centroid, and circumcenter of any triangle are collinear - that is, they always lie on the same straight line called the Euler line, named after its discoverer.
For more, and an interactive demonstration see Euler line definition.

In the case of an equilateral triangle, all four of the above centers occur at the same point.

Inradius: The radius of the incircle. The radius is given by the formula:

a is the area of the triangle. p is the perimeter of the triangle, the sum of its sides.

Radius of its circumcircle:

If you know the length (a,b,c) of the three sides of a triangle, the radius of its circumcircle is given by the formula:

If you know one side and its opposite angle

The diameter of the circumcircle is given by the formula: where a is the length of one side, and A is the angle opposite that side. This gives the diameter, so the radius is half of that.

TRIGONOMETRY

Basic formulas

cos2(t) + sin2(t) = 1

1 + tan2(t) = sec2(t)

1 + cot2(t) = cosec2(t)

Area of a triangle


Area of a triangle
 
            

The area of the triangle is a.h/2 .
But in triangle BAH, we have sin(B) = h/c .
Hence the area of the triangle is a.c.sin(B) /2.
Similarly, we have that the area of the triangle 
= b.c.sin(A) /2 = a.b.sin(C) /2


The area of a triangle ABC =(1/2) a.c.sin(B) = (1/2) b.c.sin(A) = (1/2) a.b.sin(C)




You can also use Heron's formula to calculate the area of a triangle.

 
Let s = half the circumference of the triangle = (a +b + c)/2.

The area of a triangle ABC =
     ______________________________
    V s (s - a) (s - b) (s - c)








Sine rule


          a         b        c
        ------ =  ------ = ------
        sin(A)    sin(B)   sin(C)

Let R be the radius of the circle with center O through the points A,B and C.

In any triangle ABC we have
 
          a         b        c
        ------ =  ------ = ------ = 2R
        sin(A)    sin(B)   sin(C)

Cosine rule


In any triangle ABC we have

 
        a2  = b2  + c2  - 2 b c cos(A)

        b2  = c2  + a2  - 2 c a cos(B)

        c2  = a2  + b2  - 2 a b cos(C)






Sum formulas


cos(u-v) = cos(u).cos(v)+sin(u).sin(v)

sin(u - v) = sin(u).cos(v)-cos(u).sin(v)




           tan(u) + tan(v)
tan(u+v) = -----------------
           1 - tan(u).tan(v)



t-Formulas or Half-Angle Formulas


Let t = tan(u) , then


           1 - t2
cos(2u) = ---------
           1 + t2
or
             1 -  tan2(u)
cos(2u) =   -------------
             1 + tan2 (u)

             2t
sin(2u) =  --------
            1 + t2
or
             2 tan(u)
sin(2u) =  -----------
           1 + tan2 (u)




Simpson formulas

 
                        x + y       x - y
cos(x) + cos(y) = 2 cos ------ cos -------
                          2           2

                         x + y       x - y
cos(x) - cos(y) = -2 sin ------ sin -------
                          2           2

                        x + y       x - y
sin(x) + sin(y) = 2 sin ------ cos -------
                          2           2

                        x + y       x - y
sin(x) - sin(y) = 2 cos ------ sin -------
                          2           2