**Let
Us Consider Yet another Triangle . . .**
**Final
Project**
**by**
**Robin Kirkham**

**Part A**
Consider yet
another triangle (we have not considered enough triangles) ABC.
Select a point P inside the triangle and observe the lines segments
AP, BP, and CP extended to an intersection with the opposites
sides at points D, E, and F.
Let us explore
further the segments (AF)(BD)(EC) and (FB)(DC)(EA) for a variety
of triangles and change the location of point P.

This exploration
of the segments of each of the side of the triangle
(no matter the
length - as changed by the position of P and the movement of the
vertex)
the cooresponding
segments when multiplied produced the same values in the same
areas.

**Part B**
It is not enough
to look at the simple triangle and stop there (that is no fun).
Let us consider
the ratio of
(AF)(BD)(CE)/(BF)(CD)(AE)
According to
our calculations, this ratio equals 1 everytime (even when P or
a vertex is move thus changing the length of the segments).
**Given**: triangle ABC, with -
F on line AB,
E on line AC,
D on line BC
line AD, line
BE, and line CF are concurrent at P
**Proof or
Conjecture?**
Prove:
Line XY is constructed
to be parallel to line BC below.
Connect vertices
C and B through point P making lines CX (line through P) and BY
(line through P)
On the above diagram make the following
observations:
Dreaw a line through A such that it
is parallel to line BC meeting line CP at point X and line BP
at point Y.
Notice the two triangles in yellow
AEY - CEB
therefore, (1.)
for triangles
BFC-AFX therefore, (2.)
for triangles
CDP-YAP therefore, (3.)
for triangles
BDP-XAP therefore, (4.)
from the ratios
(3.) and (4.) we get
(5.)
Now if we multiply
(1.), (2.), and (5.) providing the followowing:
and then you can show the following:

Part 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?
In the above
triangle ABJ and DFK areas are compared. Notice that the ratio
of the two areas is greater than 4.
Below the point
P has been adjusted and the two areas are compared and again the
ratios is still greater than 4.
Furthermore,
the third diagram (above) has moved point P yet again to the centroid
and still the areas are compared and are a ratio of 4. No matter
what location point P is at within the triangle the ratio of the
two areas remains at a minimum of 4.
Furthermore,
the third diagram (above) has moved point P yet again to the centroid
and still the areas are compared and are a ratio of 4.
Conclusion:
No matter what
location point P is at within the triangle the ratio of the two
areas remains at a minimum of 4.
As we observed
in assignment 4 (centers of a triangle) the CENTROID of a triangle
is the common intersection of the three medians.
This makes the
areas of the six triangles (BFP, FAP, AKP, KJP, and JDP) equal.
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