Points of concurrency in triangles

There are 3 types of concurrency in triangles.

The centre of these concurrencies are  named   Incentre, circumcentre and orthocenter.

The incenter  helps in forming the in circle in a triangle

The circumcentre is used to circumscribe a triangle.

Incentre:-

The internal bisectors of the three vertical angles are concurrent. This point of concurrency is called Incenter.

I is the Incentre,

I is found  using the formula (ax1+bx2+cx3) / (a+b+c) and  (ay1+ by2+ cy3)/( a+b+c)

 'a' is the side opposite vertex A, 'b' is the side opposite vertex B and  'c' is the side opposite vertex C.

Circumcentre, a Point of Concurrency in Triangles

The perpendicular bisectors of the sides of a triangle are  concurrent. This point of concurrency is called  Circumcenter, and is denoted by the letter S.

The circumcenter is the centre of the circle passing through the 3 vertices  A, B and C

To find S first we find the midpoint of one side of the triangle.  Let us find the midpoint of AB.

Then let us find the slope of AB.

Using slope - one point formula, we can find the circumcenter.

Orthocentre, a Point of Concurrency in Triangles

The altitudes of a triangle are concurrent. This point of concurrency is called orthocenter.  Its symbol is O.

Orthocenter is also found using slope of altitude, one point(coordinate) formula.

For the 2 altitudes we find the equations and then solve the equations.

we must find slope of AD and  use the coordinates of A to find the equation of AD.

Then we must find slope of BE , use the coordinates of B to find the equation of BE'

The solve both equations o AD and BE to get the orthocenter O.