46 CHAPTER 1 Advanced Euclidean Geometry
- Given 4 ABC, let Obe its orthocenter. LetC be the nine-point
circle of 4 ABC, and letC′be the circumcenter of 4 ABC. Show
thatCbisects any line segment drawn fromOtoC′.
1.6 Mass point geometry
Mass point geometry is a powerful and useful viewpoint particularly
well suited to proving results about ratios—especially of line segments.
This is often the province of the Ceva and Menelaus theorems, but, as
we’ll see, the present approach is both easier and more intuitive.
Before getting to the definitions,
the following problem might help
us fix our ideas. Namely, con-
sider 4 ABC with Cevians [AD]
and [CE] as indicated to the right.
Assume that we have ratios BE :
EA= 3 : 4 andCD:DB= 2 : 5.
Compute the ratiosEF :FC and
DF :FA.
Both of the above ratios can be computed fairly easily using the con-
verse to Menalaus’ theorem. First consider 4 CBE. From the converse
to Menelaus’ theorem, we have, sinceA, F, andD are colinear, that
(ignoring the minus sign)
1 =
2
5
×
7
4
×
EF
FC
,
forcingEF :FC = 10 : 7.
Next consider 4 ABD. Since the points E, F, andC are colinear,
we have that (again ignoring the minus sign)