Let I, J, K, and L be the incenter and the three excenters of triangle ABC.

What is the center of gravity of these four points?

(In reply to re: ???? by Charlie)

As the definition of "center of mass" (geometric centroid) applies to a two-dimensional planar lamina or three-dimensional solid and a point has no dimension, I was under the assumption that the "center of mass" was in reference to the the two-dimensional planar laminas of the incircle and excircles that identify the incenter and three excenters of triangle ABC. 

Of course these points are the "center of mass" of each respective circle, but these circles would not to be of equal "mass". Only in an equilateral triangle would the three excircles have the same "mass" (the incircle, even though being of different "mass", would have no effect of the position of the "center of mass" as its center would be the "center of mass"). 

Each triangle has three excircles whose radii are determined by the following equations where a, b and c are the three side lengths of the triangle ABC tangent to excircles J, K, and L and s is the semi-perimeter of triangle ABC {s = (a+b+c)/2)}:

R_J = 2*SQRT(s*[s - a]*[s - b]*[s - c]) / (-a + b + c)
R_K = 2*SQRT(s*[s - a]*[s - b]*[s - c]) / (a - b + c)
R_L = 2*SQRT(s*[s - a]*[s - b]*[s - c]) / (a + b - c)

The radius of incircle I of triangle ABC is given by the equation:

R_I = 2*SQRT(s*[s - a]*[s - b]*[s - c]) / (a + b + c)

The area ("mass") M_x of the incircle and the three excircles are as follows:

M_I = 4*pi*SQRT(s*[ s - a]*[s - b]*[s - c]) / (a + b + c)
M_J = 4*pi*SQRT(s*[s - a]*[s - b]*[s - c]) / (-a + b + c)
M_K = 4*pi*SQRT(s*[s - a]*[s - b]*[s - c]) / (a - b + c)
M_L = 4*pi*SQRT(s*[s - a]*[s - b]*[s - c]) / (a + b - c)

The "center of gravity", then, can be found by finding the barycenter between I and J => B_IJ, then between B_IJ and K  => B_IJK, and, finally, between L and B_IJK.

The barycenter is the point between two "bodies" where they balance each other, and the equation to find the distance, d, from the center of the first "body" to the barycenter is:
d = x / (1 + M₁ / M₂), where x is the distance between the two "centers" and M₁ and M₂ are the "masses" of the two bodies.

Edited on February 8, 2008, 4:01 am

Posted by Dej Mar on 2008-02-07 00:32:48