Advanced High-School Mathematics

48 CHAPTER 1 Advanced Euclidean Geometry

Applying the converse to Menelaus’ theorem to the triangle 4 PQS, we
have, sinceT, W, andRare colinear, that (ignore the minus sign)

1 =

PT

TQ

×

QR

RS

×

SW

WP

=

y

x

×

y+z

y

×

SW

WP

.

This implies thatPW :WS= (y+z) :x, which implies that

(x+y+z)W =xP+ (y+z)S=xP+ (yQ+zR).

Similarly, by applying the converse of Menelaus to 4 QRT, we have
that (x+y+z)W = (x+y)T+zR= (xP+yQ) +zR, and we’re done,
since we have proved that

xP+ (yQ+ZR) = (x+y+z)W = (xP+yQ) +zR.

The point of all this is that given mass pointsxP, yQ, andzR, we
may unambiguously denote the “center of mass” of these points by
writingxP+yQ+zR.

Let’s return one more time to the example introduced at the begin-
ning of this section. The figure below depicts the relevant information.
Notice that the assigments of masses toA, B,andC are uniquely de-
termined up to a nonzero multiple.