If and are the vertices of triangle , whose sides are of lengths respectively, then,
(a) ( To A)
(b) (To B)
(c) (To C)
(i) Incenter divides the angle bisectors in ratio, and
(ii) Incenter and Excenter are harmonic conjugate of each other wrto the angle bisector on which they lie.
(iii) Centroid divides the (‘ Euler line’ ) joining orthocentre and Circumcentre in the ratio .
(iv) In an Isoscles triangle and lie on the same line and in an equilateral triangle, all these four points coincide.
(v) In an right angled triangle orthocentre is at right angled vertex and circumcentre is midpoint of hypotenuse.
(vi) In case of an obtuse angled triangle circumcentre and orthocentre both are outside the triangle.
Figure for (iv)
Figure for (v)
The internal angle bisector of one angle and the external angle bisector of each other two angles of a triangle meet at a point is known as Excentre.