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Incenter of a triangle (Coordinate Geometry)
 
Given the coordinates of the three vertices of a triangle ABC,
the coordinates of the incenter O are
where:
Ax and Ay are the x and y coordinates of the point A etc..
a, b and c are the side lengths opposite vertex A, B and C
p is perimeter of the triangle (a+b+c)
Try this Drag any point A,B,C. The incenter O of the triangle ABC is continuously recalculated using the above formula. You can also drag the origin point at (0,0).

Recall that the incenter of a triangle is the point where the triangle's three angle bisectors intersect. It is also the center of the triangle's incircle.

The coordinates of the incenter are the weighted average of the coordinates of the vertices, where the weights are the lengths of the corresponding sides.

The formula first requires you calculate the three side lengths of the triangle. To do this use the method described in Distance between two points. Once you know the three side lengths, you calculate the perimeter as the sum of these three lengths.

Calculator

  x y
A  
B  
C  
Incenter
 
 
 

Use the calculator on the right to calculate coordinates of the incenter of the triangle ABC.

Enter the x,y coordinates of each vertex, in any order.

Things to try

  1. In the diagram at the top of the page, Drag the points A, B or C around and notice how the incenter moves and the coordinates are calculated. Try points that are negative in x and y. You can drag the origin point to move the axes.
  2. Click "hide details". Drag the triangle to some random new shape. Calculate the incenter position then click "show details" to see if you got it right.
Once you have done the above, you can click on "print" and it will print the diagram exactly as you set it.

Limitations

In the interest of clarity in the applet above, the coordinates are rounded off to integers. This can cause calculations to be slightly off.

For more see Teaching Notes

Other Coordinate Geometry topics