Mathematics CENTROID, INCENTRE, CIRCUMCENTRE ORTHOCENTRE, AND EXCENTRE OF A TRIANGLE
Please Wait... While Loading Full Video

CENTROID, INCENTRE, CIRCUMCENTRE ORTHOCENTRE, AND EXCENTRE OF A TRIANGLE

Centroid (`G`) of Triangle

`Concept :`

The point of intersection of the medians of a triangle is called the Centroid of the triangle.

Median is the line joining `A` with middle point of `BC (D).` `( AD` , `BE` and `CF` are Medians in the image )

Medians are concurrent and point of intersection of Medians divides median in ratio of `2 : 1`

The co-ordinates of the centroid of the triangle whose vertices are `(x_1, y_1 ), (x_2, y_2)` and `(x_3, y_3)` is

`((x_1 +x_2 +x_3)/3, (y_1 +y_2 +y_3)/3)`

Please see the image,

Let `A= (x_1, y_1), B = (x_2, y_2 )` and `C = (x_3, y_3) .`


Then the co-ordinates of `D` will be `D = ((x_2 +x_3)/2, (y_2 +y_3)/2)`

The co-ordinate of a point dividing `AD` in the ratio `2 : 1` are

`((2((x_2+x_3)/2)+1.x_1)/(2+1), (2((y_2+y_3)/2)+1.y_1)/(2+1))

= ((x_1+x_2+x_3)/3, (y_1+y_2+y_3)/3)`

Similarly the co-ordinate of a point dividing `BE, CF` in the ratio `2 : 1` will also be same due to symmetry.

Hence medians of a triangle are concurrent and the co-ordinates of the centroid are

`((x_1+x_2+x_3)/3, (y_1+y_2+y_3)/3)`

Fundas :
`=>` The centroid is always located inside the triangle.
`=>` The centroid divides each median in a ratio of `2 : 1.`
`=>` Each median divides a triangle into the two triangles of equal areas.
`=>` Centroid of a triangle joining the middle points of a triangle is same as that of original triangle.
`=>` The area of `DeltaGBC, DeltaGCA, DeltaGAB` are equal.
`=>` Centroid is also centre of gravity of triangle
`=>` To find the centroid of any other shape, find the centre of gravity of that image,
e.g centroid of a circle and rectangle are centre and point of intersection of diagonals respectively.



Incentre (`I`) of Triangle

`Concept :` The point of intersection of internal angle bisectors of triangle is called the incentre of the triangle.

Incenter is equidistant from all the sides of the triangle.

The biggest circle that can be drawn inside a triangle is the incircle. The incircle will touch all sides of the triangle. The radius of the circle is the length of the perpendicular line drawn from the Incenter to any side.


The co-ordinates of the incentre of the triangle whose vertices are `(x_1, y_1 ), (x_2, y_2)` and `(x_3, y_3)` are

`((ax_1 +bx_2 +cx_3)/(a+b+c), (ay_1 +by_2 +cy_3)/(a+b+c))`

Please see the image,

`(BD) / (DC) = (AB) / (AC) = c/b`

Hence `D` divides `BC` in the ratio `c : b.`
Similarly `E` divides `AC` in the ratio `c : a`.


Fundas :
`=>` The incentre is always located inside the triangle.
`=>` Internal angle bisector `AD` divides the base `BC` in the ratio of the sides containing the angle i.e. `BD: DC = c : b`.
`=>` lncentre `I` divides `AD` in the ratio `AB : BD`.
`=>` Incentre divedes each angel into to equal half angels.
`=>` `AI : ID = (b + c):a`

Circumcentre (`C`) of a Triangle

`Concept :` The circumcentre of a triangle is the point of intersection of the perpendicular bisectors of the sides of a triangle.

The circumcentre of a triangle is the point in the plane equidistant from the three vertices of the triangle.

The co-ordinates of the Circumcentre of the triangle whose vertices are `(x_1, y_1 ), (x_2, y_2)` and `(x_3, y_3)` are

`((ax_1cosA +bx_2cosB +cx_3cosC)/(acosA+bcosB+c cosC), `
`(ay_1cosA +by_2cosB +cy_3cosC)/(acosA+bcosB+c cosC))`



Fundas :
`=>` In case of acute angle triangle circumcentre lies inside the triangle.
`=>` In case of right angle triangle it lies on mid-point of hypotenuse.
`=>` in case of obruse angle triangle, it lies outside the triangle.

Never use the above formula to find the circumcentre of triangle.

Use following Method:

The steps to find the circumcenter of a triangle:

`Step 1:` Find and Calculate the midpoint of given coordinates or midpoints `(AB, AC, BC)`.
`Step 2:` Calculate the slope of the particular line.
`Step 3:` By using the midpoint and the slope, find out the equation of line `(y-y_1) = m (x-x_1)`
`Step 4:` Find out the other line of equation in the similar manner.
`Step 5:` Solve the two bisector equation by finding out the intersection point.
`Step 6:` Calculated intersection point will be the circumcenter of the given triangle.

Second Method:

Circumcenter has all three vertices at equal distance as the line joining the circumcenter to vertices is radius of circumcircle. If P is the circumcenter of triangle ABC then

`PA=PB=PC`

This will give us two equations in two variables.

Orthocentre (`H`) of Triangle

`Concept :` The orthocentre of a triangle is the point of intersection of altitudes ( the lines through the vertices and perpendicular to opposite sides).

The co-ordinates of the Circumcentre of the triangle whose vertices are `(x_1, y_1 ), (x_2, y_2)` and `(x_3, y_3)` are

`((ax_1secA +bx_2secB +cx_3secC)/(asecA+bsecB+csecC),`
` (ay_1secA +by_2secB +cy_3secC)/(asecA+bsecB+csecC))`


Fundas :
`=>` In case of acute angle triangle orthocentre lies inside the triangle.
`=>` In case of right angle triangle orthocentre lies at the vertex where it is right angled.
`=>` In case of obtuse triangle orthocentre lies outside the triangle.

Never use the above formula to find the Orthocentre of triangle.

Use following Method:

`Step 1:` Calculate the slope of the sides of the triangle. The formula to calculate the slope is given as,
`text(Slope of a Line)`` =(y_2-y_1)/(x_2-x_1)`.

`Step 2:` Calculate the perpendicular slope of the sides of the triangle. It gives us the slope of the altitudes of the triangle. The formula to calculate the perpendicular slope is given as,
`text(Perpendicular Slope of a Line)`` =(-1)/(text (Slope of a Line))`

`Step 3:` Then by using point slope form, calculate the equation for the altitudes with their respective coordinates. The point slope formula is given as, `y-y_1=m(x-x_1)`.

`Step 4:` Finally by solving any two altitude equation, we can get the orthocenter of the triangle.



Excentres (`I_1, I_2, I_3`) of triangle

`Concept:` Excentre is a point of concurrency of two external angle bisectors and one interior angle bisector.
or
Centre of a circle (excircle) which touches all the sides of the triangle externally.

There are three ex centres with respect to a given triangle.

See figure

`I_1` : Coordinates of center of ex-circle opposite to vertex `A`

`I_1(x,y)=((-ax_1+bx_2+cx_3)/(-a+b+c),(-ay_1+by_2+cy_3)/(-a+b+c))`


`I_2` : Coordinates of center of ex-circle opposite to vertex `B`

`I_2(x,y)=((ax_1−bx_2+cx_3)/(a−b+c),(ay_1−by_2+cy_3)/(a−b+c))`


`I_3` : Coordinates of center of ex-circle opposite to vertex `A`


`I_3(x,y)=((ax_1+bx_2−cx_3)/(a+b−c),(ay_1+by_2−cy_3)/(a+b−c))`


Relative Position of Centroid, Circumcentre, Orthocentre and Incentre

`1.` For any triangle Orthocentre `(0)`, Centroid `(G)`, Circumcentre `(C)` are collinear and centroid divides orthocentre and circumcentre in the ratio `2 : 1` internally. This line is called Euler's line.

`2.` For isosceles triangle centroid, circumcentre, orthocentre and incentre are collinear. i.e Incemtre lies on Euler's line..

`3.` For equilateral `Delta` centroid, circumcentre, orthocentre and incentre coincide.

Centroid, Circumcentre, Orthocentre and Incentre in case of right angled triangle

`1.` Orthocentre of a right angled triangle is the vertex, which is right - angle.
`2.` Circumcentre of a right angled triangle is mid-pint of hypotenuse.
`3.` Centroid and incentre of a right angled triangle have to be determined as for any other triangle.

Another way to remember definition of various points.

Remember two points based on middle point of the base. ( Centroid and Circumcentre)
Remember other two points based on vertex. ( Incentre, orthocentre )

Centroid is point of intersection of lines which are drawn from middle point of base to vertex.
Circumentre is point of intersection of perpendicular's drawn from middle point of base.
Orthocentre is Point of intersection of perpendiculars drawn from vertex on to base.
Incentre is point of intersection of angle bisector of vertices.


Incentre Vs Circumcentre

Incentre is equidistant from the sides of triangle, whereas Circumcentre is equidistant from 3 vertices of a triangle.

 
SiteLock