Tiffany N. KeysŐ Final Assignment:

CevaŐs
Thereom

Ceva's
Theorem. Consider any triangle ABC. Select a point P inside the triangle and
draw lines AP, BP, and CP extended to their intersections with the opposite
sides in points D, E, and F respectively.

1.
Explore (AF)(BD)(EC) and
(FB)(DC)(EA) for various triangles and various locations of P.

2.
Conjecture? **Prove it!** (you may need draw
some parallel lines to produce some similar triangles, OR, you made need to
consider areas of triangles within the figure) Also, it probably helps to
consider the ratio

__(AF)(BD)(CE)__

(BF)(CD)(AE)

3.
In your write-up, after the
proof, you might want to discuss how this theorem could be used for proving
concurrency of the medians (if P is the centroid), the lines of the altitudes
(if P is the orthocenter), the bisectors of the angles (if P is the incenter), or
the perpendicular bisectors of the sides (if P is the circumcenter).
Concurrency of other special points?

4.
Explore a generalization of the
result (using lines rather than segments to construct ABC) so that point P can
be **outside**
the triangle. Show a working GSP sketch.

LetŐs
explore (AF)(BD)(EC) and (FB)(DC)(EA) for various triangles and various
locations of P.

OBSERVATIONS:

á
The
ratios of (AF)(BD)(EC) and (FB)(DC)(EA) from this triangle with P as the
centroid turned out to be equal .

á
From
looking at this triangle, it was assumed that no matter where P is located, the
ratios will always be the same and remain equal.

** **

**Ceva's
Theorem states:**

If
the points F, E, and D are on the sides AB, AC and BC of a triangle then the
lines AC, BE and CF are concurrent if and only if the product of the ratios

__(AF)(BD)(CE)__

(FB)(DC)(EA)

PROOF:

1. __AF__ = __D____ACP__

FB DBCP

2. __BF__ = __D____ABP__

FA DACP

3. __CE__ = __D____BCP__

EA DABP

Therefore, (__AF)(BF)(CE)__ = (__D____ACP)( ____D____ABP)( ____D____BCP)__

(FB)(FA)(EA) (DBCP)( DACP)( DABP)