The Art and Craft of Problem Solving

288 CHAPTER 8 GEOMETRY FOR AMERICANS

It should be possible, with the techniques we have acquired, to determine when

three cevians are concurrent. A very elegant answer was discovered in the 17 th century

by Giovanni Ceva (now you know where "cevian" came from).

Example 8.4.3 Ceva's Theorem. The cevians AD, BE, CF are concurrent if and only

if

c

AF BD CE

FB DC EA =

1.

A

B

Proo!' First we need to recall a simple algebra lemma involving proportions:

Suppose that

Then it is also true that

x a

y b

x x+a x-a

y y+b y-b'

This lemma is very easy to check, even though it seems surprising at first. We are

conditioned to expect a fraction to stay the same when we mUltiply the numerator and

denominator by the same number, but we expect trouble when we add or subtract from

the numerator and denominator.

With this lemma, Ceva's theorem is pretty simple. Suppose that the cevians are

concurrent at G. Converting to areas, as labeled above, we have

AF y x+y+z x+z

FB u+v+w u+v

,

w

BD v y+w+v y+w --

,

DC u x+z+u x+z

CE z z+u+v u+v

EA x+y+w y+w

,

x

and the product of these expressions is clearly 1. We leave the converse (if the product

is 1, then the cevians are concurrent) as an exercise. _