Complete each of the following parts. Part One is merely finishing off the write-ups for the semester. Part Two should be a new web page item. Part Three is the course evaluation.
Review your Write-ups 1 to 12. Revise as you wish to reach the point where you feel that, collectively, the 12 write-ups represent your best work for the course. They are an elctronic portofolio of your work.
Complete a Write-up on your Web Page for the following investigation. This should be individual work.
A. 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.
Explore (AF)(BD)(EC) and (FB)(DC)(EA) for various triangles and various locations of P.
B. Conjecture? Prove it! (you may need draw some parallel lines to produce some similar triangles) Also, it probably helps to consider the ratio
Can the result be generalized (using lines rather than segments to construct ABC) so that point P can be outside the triangle? Show a working GSP sketch.
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?