Question

Given A (3,2) , B (4,3) , C(6,1) are the vertices of a triangle find incentre , orthocentre,circumcentre ?

Answer 1

@mukushla @Mashy @.Sam. @ajprincess

Answer 2

@Diwakar

Answer 3

I am not sure if you tackle this problem in case of a vector geometry class, or just from the definition of lines and intersection. But I would start with the incenter, if I don't confuse the english terms then there's a neat formula for that:
I=(1/3)*(A+B+C)

Answer 4

For the remaining problems it seems to me like you want to define some lines and intersect them, you basically need the normals of each line. So maybe this helps you:

http://www.mathopenref.com/trianglecircumcenter.html

http://www.mathopenref.com/triangleorthocenter.html

Answer 5

For orthocenter
We now it is the point of intersection of altitudes.
AB and BE are the two altitudes. Since AD is perpendicular to BC,we can find slope of AD as we can find the slope of BC. With the slope of AD and point A lying on AD, we can write its equation. Similarly write the equation of BE(which is perpendicular to AC and has point B on it). Solve the two equations to find the point of intersection (orthocenter)

Answer 6

For circumcenter
It is the point of intersection of perpendicular bisectors of sides. The method is only slightly different from previous. We can find points D and E which are mid points of AB and BC. Also like before we can find the slopes of these bisectors. Then write their equations and solve for their point of intersection.

Answer 7

Incenter is little less obvious to me. The main problem is to find the slope of the internel angle bisectors of the triangle. Once we get that it is easy to write eqn of angle bisectors and solve them for incenter. What we can do is to find the angle made by side AB with x axis which is related to the slope as the tangent of this angle. Similarly find the angle which is made by BC with x axis. With this information you may then find the angle made by bisector of angle B with x axis (it will be the average of these two angles). Then you may find its eqn. Similarly find the eqn of the bisector of angle C (or A). Solve them and find incenter.

Answer 8

@Spacelimbus
I think the formula you have written above is for centroid of a triangle.

Answer 9

You're right about that @Diwakar, I should have continued reading upon the articles I have linked above.

http://www.mathopenref.com/triangleincenter.html

Answer 10

Hey i found this nifty formula for incenter on wikipedia.
http://en.wikipedia.org/wiki/Incenter#Coordinates_of_the_incenter