Here, you will learn formula for circumcenter of a triangle and formula for incenter of a triangle.
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Formula for Circumcenter of a Triangle
It is the point of intersection of perpendicular bisectors of the sides of a triangle. If O is the circumcenter of any triangle ABC, then \(OA^2\) = \(OB^2\) = \(OC^2\). Also it is the center of circle touching all the vertices of a triangle.
Note :
(i)ย If the triangle is right angled, then its circumcenter is the mid-point of the hypotenuse.
(iii)ย The Co-ordinates of circumcenter is :
Circumcenter = (\(x_1sin2A+x_2sin2B+x_3sin2C\over {sin2A+sin2B+sin2C}\),\(y_1sin2A+y_2sin2B+y_3sin2C\over {sin2A+sin2B+sin2C}\))
Incenter of a Triangle
The incenter is the point of intersection of internal bisectors of the angles of a triangle. Also it is a centre of the circle touching all the sides of the triangle.
Co-ordinates of incenter I is (\(ax_1+bx_2+cx_3\over {a+b+c}\),\(ay_1+by_2+cy_3\over {a+b+c}\)) where a,b,c are the sides of the triangle ABC.
Note :
(i)ย Angle bisector divides the opposite sides in the ratio of remaining sides e.g. \(BD\over DC\) = \(AB\over AC\) = \(c\over b\)
(ii)ย Incenter divides the angle bisectors in the ratio (b+c) : a, (c+a) : b, (a+b) : c.
Remarks :
(i)ย If the triangle is equilateral, then centroid, incentre, orthocenter, circumcenter coincide.
(ii)ย Orthocenter, centroid and circumcenter are always collinear and centroid divides the line joining orthocenter and circumcenter in the ratio 2 : 1.
(iii)ย In an isosceles triangle centroid, incenter, orthocenter and circumcenter lie on the same line.
Hope you learnt formula for circumcenter of a triangle and formula for incenter of a triangle. To learn more practice more question and get ahead in competition. Good Luck!
