SECTION 1.6 Mass Point Geometry 51
Example 3. Prove that the angle bisectors of 4 ABCare concurrent.
Proof. Assume that AB =c, AC =b, BC = a and assign masses
a, b,andcto pointsA, B, andC, respectively. We have the following
picture:
Note that as a result of the Angle Bisector Theorem (see page 15) each
of the Cevians above are angle bisectors. Since the center of mass is on
each of these Cevians, the result follows.
The above applications have to do with Cevians. The method of
mass point geometry also can be made to apply totransversals, i.e.,
lines through a triangle not passing through any of the vertices. We
shall discuss the necessary modification (i.e., mass spltting) in the
context of the following example.
Solution. The above examples were primarily concerned with com-
puting ratios along particular Cevians. In case a transversal is in-
volved, then the method of “mass splitting” becomes useful. To best
appreciate this, recall that if in the triangle 4 ABC we assign massa
toA,btoB, andcto C, then the center of massP is located on the
intersection of the three Cevians (as depicted below). However, sup-
pose that we “split” the massbatBinto two componentsb=b 1 +b 2 ,
then the center of massP will not only lie at the intersection of the
concurrent Cevians, it will also lie on the transversal [XZ]; see below: