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

B. Conjecture? Prove it! (you may need draw some parallel lines to produce some similar triangles) Also, it probably helps to consider the ratio (AD)(EB)(CF)/(DB)(CE)(AF).

__MY
CONJECTURE:__
For any triangle ABC and for any point P, inside triangle ABC,
(AF)( BD)(EC) = (FB)(DC)(EA). Therefore (AD)(EB)(CF)/(DB)(CE)(AF)
= 1 for any triangle ABC and any point P inside of triangle ABC.
Click
here
for a GSP sketch in order to explore the construction as well
as my conjecture.

__PROOF
of MY CONJECTURE:__

Let's begin by extending segments CD, BF and AE into rays. Also, let's draw in some parallel lines that will be of use.

Now, let's tally up our pairs of similar triangles. (Let "~" mean "is similar to"). Because (Line HA) ll (Ray BF) ll (Line IC) which leads to the AAA similarity postulate, we have that:

Well, because Triangle AHD ~ Triangle BPD, we know that (AH)/(BP) = (AD)/(BD). Also, because Triangle BPE ~ Triangle ICE, we know that (BE)/(CE) = (BP)/(IC). Well, if X and Y are equivalent and you increase both of them by an equivalent amount, c, then cX =cY. In the same way,

Because Triangle AHC ~ Triangle PFC, we know that (PF)/(AH) = (CF)/(AC). Also, because Triangle ICA ~ Triangle PFA, we know that (IC)/(PF) = (AC)/(AF). Therefore,

Remember: (AD)/(BD) *(BE)/(CE) = (AH)/(IC). And we now have that: (IC)/(AH) = (CF)/(AF).

Cancelling like terms we get:

Multiplying both sides by (BD)(CE)(AF), we get:

Can the result be generalized (using lines rather than segments to construct ABC) so that point P can be outside the triangle? Yes. Here is a sketch and measurements showing that the result does indeed generalize.

To satisfy any dounts, click here for a GSP sketch of the above construction. You will be able to see the measurements stay the same even as you manipulate triangle ABC and the point P.

Let the area of Triangle ABC be represented by A(abc) and let the area of Triangle DEF be represented by A(def). I want to show that A(abc)/A(def) is always greater than or equal to 4/1. This means that A(def) is never larger than 1/4 of the area of Triangle ABC.

Let: segment BC = a, segment AC = b, and segment AB = c. Let: segment FE = d, segment DF = e, and segment DE = F. Let a,b,c,d,e and f be non-zero constants.

The area of Triangle ABC = (1/2)ab[sin(angle ACB)].

The area of Triangle DEF = ( 1/2 )ef[sin(angle DFE)].

**When
is A(abc)/A(def) = 4/1 exactly?** I propose that it is exactly
when D, E and F are midpoints of AB, BC and AC respectively.

By the Midpoint Connector Therom, we get the following relationships between Triangle ABC and Triangle DEF (these whole only when D, E and F are midpoints of AB, BC and Ac respectively):

Therefore, when D, E and F are midpoints of Triangle ABC, we get that

Because Angle DFE = Angle ACB, we can substitute and get:

Therefore, the ratio of A(abc)/A(def) = A(abc)/(1/4)[A(abc)] = 1/(1/4) = 4. Thus, my proposition is correct. Our ratio equal 4 exactly when D, E and F are midpoints of Triangle ABC.

**Is
A(abc)/A(def) always greater than or equal to 4?(i.e., is A(def) never larger
than 1/4 of A(abc)?) **Yes. Click
here
for a GSP sketch (similar to the one below) and an animation showing
that the ratio: A(abc)/A(def) is in fact always greater than or
equal to 4.

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