Unit-I: Sets and Functions
Unit-II: Algebra
Unit-III: Coordinate Geometry
Unit-IV: Calculus
Unit-V: Mathematical Reasoning
Unit-VI: Statistics and Probability

Formula to find coordinates

If   A (x_1,y_1) , B (x_2,y_2)   and  C (x_3,y_3)   are the vertices of triangle  ABC  , whose sides  BC, CA, AB   are of lengths  a, b ,c   respectively, then,

(i) Centroid  ( G )=\left( \dfrac{x_1+x_2+x_3}{3},\dfrac{y_1+y_2+y_3}{3} \right)

(ii)  Incenter  (I) =\left( \dfrac{ax_1+bx_2+cx_3}{a+b+c},\dfrac{ay_1+by_2+cy_3}{a+b+c} \right)

(iii) Excenter

(a) ( To A)  (I_1) =\left( \dfrac{-ax_1+bx_2+cx_3}{-a+b+c},\dfrac{-ay_1+by_2+cy_3}{-a+b+c} \right)

(b) (To B)  (I_2) =\left( \dfrac{ax_1-bx_2+cx_3}{a-b+c},\dfrac{ay_1-by_2+cy_3}{a-b+c} \right)

(c) (To C)  (I_3) =\left( \dfrac{ax_1+bx_2-cx_3}{a+b-c},\dfrac{ay_1+by_2-cy_3}{a+b-c} \right)

Notes:

(i) Incenter divides the angle bisectors in ratio,   (b+c) : a ; (c+a) : b  and  (a+b) : c

(ii) Incenter and Excenter are harmonic conjugate of each other wrto the angle bisector on which they lie.

(iii) Centroid ( G ) divides the (‘ Euler line’ ) joining orthocentre  ( H ) and  Circumcentre  ( O ) in the  ratio  2 : 1 .

(iv) In an Isoscles triangle   G, O ,I and  H  lie on the same line and in an equilateral triangle, all these four points coincide.

(v) In an right angled triangle orthocentre ( H ) is at right angled vertex and circumcentre  ( O )   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)

Excenter  (I^1)

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.

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