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.

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

When P is on the side of the triangle ABC, a line segment connecting
the vertex and the opposite side divides

the triangle ABC into two. But if P is inside the triangle
ABC six triangles are formed. the line segment

AF, FB, Bd, Dc, CE, and EA intersect with two line segments
from the vertex P to form the triangle

APF. APE, FPB, EPC BPD and CPD.

When the vertex of the triangle ABC is dragged, the inside
triangles change shapes.

Similar pairs of triangles are observed.

__Conjecture:__

(AF)(BD)(EC) = (FB)(DC)(EA)

__Prove:__

To prove the above conjecture, let examine the similarity of the
inside triangles of the construction below.

Let the triangle ABC an isosceles triangle. If P is on the segment
AD which bisect the segment BC, then the triangle

ABD and ACD are congruent by SAS theorem.

So is the triangle BEC and BEA.

If two triangle are similar, then the measures of the corresponding
altitude are proportional to the measure of the corresponding
sides.

Based on the above theorem I can conclude the triangle BPD is
similar ot the triangle

CPD, the triangle BPF is similar ot the triangle CPF, and the
triangle APF is similar ot the

triangle APE,

Click **here**
to see the GSP of the above construction.

Another aspect to prove the similarity between the pairs
of triangles is to examine their

areas.

The ratio: (AF)(BD)(EC) / (FB)(DC)(EA) =1

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.

the reason why the ratio is always greater than 4 is due to the
fact that the largest area of

the triangle DEF is one fourth of the triangle ABC.

The triangle DEF reaches the largest possible area when DEF is
an isosceles triangle

and the remaining triangles are also isosceles triangles.

Click **here**
to see the GSP of the above construction.