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Wednesday, November 9, 2016

Centroid, circumcentre, orthocentre, incentre of triangle

Centroid:
  • The centroid of a triangle is the point of intersection of medians. It divides medians in 2 : 1 ratio.
  • If  A(x1, y1), B(x2, y2), C(x3, y3) are vertices of triangle ABC, then coordinates of centroid is
    G = [(x1 + x2 + x3)/ 3, (y1 + y2 + y3)/ 3]
In center: 
  • Point of intersection of angular bisectors
  • Coordinates of
    I = [(ax1 + bx2 + cx3)/ a + b + c, (ay1 + by2 + cy3)/ a + b + c]
  • Where a, b, c are sides of triangle ABC.
Circumcentre:
  • Point of intersection of perpendicular bisectors.
  • Co-ordinates of circumcentre O is
    O = [(x1sin2A+x2sin2B+x3sin2c)/ sin2A+sin2B+sin2c, (y1sin2A+y2sin2B+y3sin2c)/ sin2A+sin2B+sin2c]
Orthocenter:
  • Point of intersection of altitudes of triangle ABC.
  • Coordinates of orthocenter H is
    H = [(x1tanA + x2tanB + x3tanc)/ tanA+tanB+tanc, (y1tanA + y2tanB + y3tanc)/ tanA+tanB+tanc]
Note:
  • Orthocenter of a right angled triangle is at its vertex forming the right angle.
  • The orthocenter H, circumcentre O and centroid G of a triangle are collinear and G divides H, O in ratio 2 : 1 i.e., HG : OG = 2: 1.
  • Circumcentre of a right angled triangle is mid-point of hypotenuse.
Nine Point Circle:
  • Let ABC be triangle such that AD, BE and CF are its altitudes, H, I, J are midpoints of line segments of sides BC, CA, AB respectively; K, L, M are midpoints of joining orthocenter (O) to angular points A, B, C. 
  • These 9 points (D, E, F, H, I, J, K, L, M) are concyclic and the circle passing through nine points is nine point circle and Center is known as nine- point center.


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