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:




Letís 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.






By simplifying I get:



BD*AF*CE††††† = 1