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.

By using GSP, let's first draw an arbitrary triangle having conditions as mentioned at question.

To find any connectioon between (AF)(BD)(EC) and (FB)(DC)(EA),

I calculate each segments' length and multiply.

Interestingly, those two multiplications come out same result.

so I guess **(AD)(BD)(EC)
= (FB)(DC)(EA)**. ( to see it always
satisfied, click here to see GSP tool)

*** **For Various Triangles *****

1) **an isosceles triangle**

2) **a right triangle**

**3) an equilateral triangle**

***** let's move around the point P!!! *****

When P is a ** Centroid**,

When P is an ** Incenter**,

When P is a ** Circumcenter**,

When P is a *Orthocenter,*

When P is at the outside of the triangle,

**B**. Conjecture? **Prove it!** (you may need draw some
parallel lines to produce some similar triangles) Also, it probably
helps to consider the ratio

Can the result be generalized (using lines
rather than segments to construct ABC) so that point P can be
**outside** the triangle? Show a working GSP sketch.

As I conjectured at **A,
(AD)(BD)(EC) = (FB)(DC)(EA) **always satisfied for any triangles
and any locations of P.

Now let's think about

**------------*,**

By using ###((AF)/(BF) = (FH)/(CF) = (AH)/(BC)), ####((CE)/(AE) = (CB)/(AG) = (EB)/(EG))

Substitute (AH)/(BC) instead of (AF)/(BF), and (CB)/(AG) instead of (CE)/(AE) .

Then we will get the equation as this;

(AH)(BD)(CB)/(BC)(CD)(AG).

By cancelling out same factor (BC), (AH)(BD)/(CD)(AG) is left.

Then let's subtitute (DP)/(AP) instead of (BD)/(AG), and (AP)/(DP) instead of (AH)/(CD).

Likewise, by cancelling out, (DP)(AP)/(AP)(DP) is 1.

Thus, **=** 1 ;

(AF)(BD)(CE) = (BF)(CD)(AE) is proved.

**C**. 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. When
is it equal to 4?

Let's see the picture below,

We can easily see through the result I got from the triangle ABC; area of the triangle ABC is greater than the are of the inner triangle DEF.

Let's find the case when the area of ABC is exaclty 4 times the area of EFC.

Concider that the point P is centroid,

Now, it is obvious to see that the area of the triangle ABC is exactly 4 times larger than the triangle DEF.

When point P is a centroid, segment EF and BG is parallel, so the length EF is 1/2 of the length BC.

So the area of is 1/4th of the original triangle,

since and is having same area,

as a result, tringle ABC = 4 * DEF.