Final Assignment
By
Cara
Haskins
The Final Assignment will begin with any triangle ABC. P is any point inside the triangle ABC, and EFD are the intersecting points of the segments drawn from the vertices of triangle ABC going through any point P.
The above
illustrations demonstrate the relationships between line segments AF, BD, and
CE in proportion to line segments FB, DC, and EA. Notice that the size of the triangle may change, and the
position of point P inside the triangle may change, and yet through it all the
product of the orange line segments will always equal the product of the blue
line segments. We may then
conclude from this that the ratio (AF)(BD)(CE)/(BF)(CD)(AE) will always equal
1. You donÕt believe me? Click here to try
for yourself. This is known as CevaÕs Theorem, which states that in a
triangle ABC, three lines AD, BE, and CF intersect at a single point P if and
only if (AF)(BD)(CE)/(FB)(DC)(EA)=1
Now this
must be proved.
ÒProofs are to mathematics what spelling (or even calligraphy)
is to poetry. Mathematical works do consist of proofs, just as poems do consist
of characters.Ó
Vladimir Arnold
Okay, so I
did not come up with this completely on my own. I must give credit to www.cut-the-knot.org/Generalization/ceva.shtml.
First
extend the line segments beyond the triangle until they meet the parallel line,
which was made through point B parallel to AC. Now we see several pairs of similar triangles: BJD and CAD,
BFI and CAF, BIP and CEP, AEP and BJP.
From the
similar triangles we conclude that
1. BD/DC =
BJ/CA
2. AF/FB =
CA/BI
3. BI/CE =
BP/EP
4. BJ/EA =
BP/EP
From the last two we conclude that
5. BI/CE = BJ/EA and therefore,
6. CE/EA = BI/BJ
Now,
multiplying #1, #2, and #6 we see:
(BD/DC)(AF/FB)(CE/EA)
= (BJ/CA)(CA/BI)(BI/BJ) = (BJ)(CA)(BI)/(CA)(BI)(BJ) = 1
Click here to see if it will always equal 1, with point
P inside or outside of the Triangle ABC.
Okay, just in case you are
looking for even more fun, let me show that when P is inside triangle ABC, the
ratio of the areas of triangle ABC and triangle DEF is always greater than or
equal to 4.
Click
here to move point P to different locations inside the triangle ABC and see
what happens to the area of the ratio.
The ratio of the areas of triangle ABC and triangle DEF equal exactly
four only when D, E, and F are the midpoints of triangle ABC and point P is the
centroid of the triangle.