Investigation of any point P
Inside a given Triangle ABC

by
Jennifer Roth

If we consider any triangle ABC, and select any point P inside the triangle, and draw lines AP, BP, and CP we get the following picture(Looking at measurements for a specific P)

For any point P, it appeas that (AF)(BD)(EC)=(FB)(DC)(EA), but can we prove that.
In other words we are trying to show that the ratio of the two equals 1.

First we need to construct a line paralled to AD through B, and then another such paralled ling through C. We get the following picture.

Triangle BDP is similar to Triangle BCI since the corresponding angles are congruent,
Triangle DPC is similar to Triangle BHC for the same reason.
Triangle HFB is similar to PFA since the alternate interior angles are congruent.
Triangle IEC is similar to PEA for the same reason.
Therefore, we get the following ratios.


Click here to view the GSP file that demonstrates this.

Click here to view the GSP file that demonstrates this.

Click here to view the GSP file that demonstrates this.

If we manipulate the following ratios

we get the following relationship.

Click here to view the GSP file that demonstrates this.
Using this relationship we get the following

or

If we construct everything using lines, we can place point P outside triangle ABC, and generalize.

Click here to view the GSP file that demonstrates this.

The ratio of the areas of ABC and DEF is always less than or equal to 4.
Click here to view the GSP file that demonstrates this.
When is the ration equal to 4?
If we construct point P as the centroid, we get the following picture.


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