Final
Assignment
By
Nikki Masson
Part
I:
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.
Now move
the point P around and explore the lengths of AF, BD, CE, EA,
FB, and DC.
A. Calculate:
B. Move
the point P and calculate the same relationship:
C. Move
the point P one more time and again calculate the relationship:
Conjecture:
When P is inside the triangle the relationship (AF*BD*CE)/(EA*FB*DC)=1.
Part
II:
Now we will
prove the conjecture we just made.
This conjecture
we just made is called Ceva's Thoerem and
it states that:
If the
points F, D and E are on the sides AB, BC and AC of a triangle
then the lines AD, BE and CF are concurrent if and only if the
product of the ratios
Proof:
1.) Extend
the lines BE and CF beyond the triangle and draw a line through
A and parallel to BC. Mark the points where the the extended lines
cross the parallel line.
2.) There
are several pairs of similiar triangles which give us the following
ratios:
a.)
EBC and EAY are similiar triangles
So
we get the ratio: (EC/EA)=(BC/YA)
b.
FBC and FAX are similiar triangles
So
we get the ratio: (FA/FB)=(AX/BC)
c.
XAP and CDP are similiar triangles
So
we get the ratio: (XA/CD)=(PA/PD)
d.
BDP and YAP are similiar triangles
So
we get the ratio: (BD/YA)=(PD/PA)
If
we mulitply the ratios together, we get:
(BD/YA)*(XA/CD)*(EC/EA)*(FA/FB)=(PD/PA)*(PA/PD)*(BC/YA)*(AX/BC)
Then
when we simplify we get:
(BD/CD)*(XA/GA)*(EC/EA)*(FA/FB)=(AX/GA)
Next
mulitple both sides by (GA/AX)
Then
our final formula is:
(BD/CD)*(EC/EA)*(FA/FB)=1!
We
have proved our conjecture.
What
would happen if P is outside the triangle? You can explore by
using the GSP
Sketch.
Part
III:
Show that when P is inside the triangle ABC, the ratio of the
areas of the triangle ABC and triangle DEF is always greater than
or equal to 4. When is it equal to 4?
![](Graphs/image25.gif)
What if
we move point P?
Let's move
point P one more time?
Now let's
explore why our conjecture that the relationship between these
two triangles is greater than or equal to 4.
The medial
triangle DFE is 1/4 of the area of the triangle ABC, so we have
a ratio of 4 to 1.
When does
the ratio equal 4 exactly? Use the link and explore the relationship
when P is on the centriod.
GSP
Sketch
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