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Final Assignment
By: Lauren Lee
For this assignment, 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.
I have used GSP to construct triangle ABC with interior point P. Now I want to explore (AF)(BD)(EC) and (FB)(DC)(EA) for various
triangles and various locations of P.
For this ABC triangle example, I used GSP to find the measurements of
the sides. In this case:
Let’s look at more examples by moving the point P inside the triangle.
From these examples we can see that (AF)(BD)(EC) and (FB)(DC)(EA) are equal. Will they always be equal no matter where
the point P lies in the interior?
I want to make the conjecture that AF*BD*EC =
FB*DC*EA. In other words,
To prove this, I will use similar triangles. My first step is to construct parallel lines. I constructed lines through B and C that are
parallel to the line AD.
By using the alternate interior angle theorem and vertical angles, I
know that triangles DPC and BMC are similar and triangles BDP and BCN are
similar. I know from the properties of
similar triangles that:
BD = DP and BC = BM
BC CN DC DP
By using the alternate interior angle theorem and vertical angles
again, I know that triangle BMF is similar to triangle APF and that triangle
CNE is similar to triangle APE. Now I
know from the properties of similar triangles:
AF = AP and CE = CN
BF BM AE AP
Now I need to multiply
my equations together.
BD*BC*AF*CE = DP*BM*AP*CN
BC*DC*BF*AE CN*DP*BM*AP
By simplifying I get:
BD*AF*CE = 1
DC*BF*AE