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.
BD*BC*AF*CE =†††† DP*BM*AP*CN
By simplifying I get:
BD*AF*CE††††† = 1