Final Assignment

Charles Meyer

For the final for this class we are asked to create a triangle ABC with an interior point P.

Next we are asked to extend lines from A to P, B to P, & C to P. We are then asked to extend those lines so that they intersect with the line segments of the triangles.

Finally we explore the products of the equations (AF)(BD)(EC) and (FB)(DC)(EA) for many different triangles.

It would appear that no matter what the shape or the location of the point P, the products of the two equations always are equal to one another. If a ratio were developed of

(AF)(BD)(EC) /(FB)(DC)(EA), it would always equal 1.00

Now, will it always equal 1, that is the proof I hope to show.

I will begin by creating parallel line segments to those of my original triangle. If I am correct, the ratio of (JC)(BH)(HA)/(AJ)(IB)(IC) should also equal 1.

Through calculation, I find that the ratio of the lengths of the parallel line segments does also give us an answer of 1.

Comparing the original to the expanded triangle, the idea of similar triangles helps prove that the ratios will always equal 1.

Now we should consider the area of the original triangle ABC and the new triangle DEF. The triangle is formed when connecting the points created on the line segments of AC, AB, & BC.

The goal is to find the ratio of the areas of ABC and DEF. I hope to show that ABC/DEF will always be 4 or greater. Below are a couple of examples of different triangles ABC & DEF and the ratios created. Note that value never goes below 4.

Finally we consider when or if, the value of the ratio will be equal to 4. The only time that the value will be 4 is when the segments of the similar triangles are parrallel to each other. Note that segment EF is parallel to BC, DF to AC, and AB to ED. An example of this is shown below.


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