### Final Project

### By

### Michael McCallum

##### Introduction:

Consider a general triangle ABC. Place a point P inside the
triangle and then draw the concurrent segments AD, BE and CF through
point P from the respective vertices to the opposite sides. We
will explore the relation between (AF)(BD)(CE) and (FB)(DC)(EA)
for various triangles and points P.

Begin with triangle ABC and point P as shown below.

Notice that (AF)(BD)(CE) = (FB)(DC)(EA). Is this always true?
Let's move point P towards side AC and see if this is still true.

Yes, the products are still equal. What about for a different
triangle ABC? Let's relocate vertex A to make a different triangle.

Again, the products are equal. Let's make a conjecture:

#### Conjecture:

Given any triangle ABC and three line segments AD, BE and CF
connecting the vertices and the opposite sides,. intersecting
at a point P, then (AF/FB)(BD/DC)(CE/EA) = 1.

#### Proof:

Extend segments BE and CF until they meet the line parallel
ot BC drawn through vertex A. Label these intersections G and
H. Then we have the following similarities by angle - angle.

AHE~CBE

AGF~BCF

AHP~BDP

AGP~CDP

Frome these we have the following ratios:

AE/EC = AH/BC;

BF/FA = BC/AG

AG/CD = AP/PD

AH/BD = AP/PD

From the last two we conclude that AG/CD = AG/BD and hence,
by multiplying both sides by CD/AH, we have AG/AH = CD/BD.

Now, from the above, we have:

(AE/EC)(BF/FA)(CD/BD) = (AH/BC)(BC/AG)(AG/AH) = 1.

Which was to be shown.

**Actually, this is just the first part of the proof of Ceva's
Theorem which states:**

*In a triagle ABC, three lines AD, CE and BF intersect
if and only if*

*(AF/FB)(BD/DC)(CE/EA) = 1.*

Is it possible to generalize the above for points P outside
the triangle? Yes. Look at the sketch below. The sketch was made
by extending all of the segments to lines and moving the point
P outside the triangle.

To satisfy yourself that this is true for any location of P,
click here to view a Geometer's
Sketch Pad sketch of the above and move point P around and observe
the results.

Part C.

**End of Proof**

**Return**