Final Assignment

By Sharren M. Thomas

THE PROBLEM:

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. CLICK HERE to Manipulate the triangle to obtain various triangles and to analyze for 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.
It appears that will be equal to 1, for various locations of P and for various triangles.

I wasn't able to prove this conjecture on my own, but the following websites gave me assistance with the Ceva Theorem.

http://www.ies.co.jp/math/java/vector/ceva/ceva.html

http://planetmath.org/encyclopedia/CevasTheorem.html

http://www.cut-the-knot.org/Generalization/ceva.shtml

http://www.math.uci.edu/~mathcirc/math194/lectures/advanced3/node2.html

Problem

Prove Ceva's Theorem. Ceva's theorem states: If the points D,E and F are on the sides AB, BC and AC of a triangle then the lines AE, BF and CD are concurrent if and only if the product of the ratios .

Answer

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?

The ratio appears to be equal to 4 when points D, E, and F are the midpoints of the sides ABC. CLICK HERE TO INVESTIGATE WHEN THE RATIO IS GREATER THAN 4.

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