Below is a picture that helps to show what the question is asking.


I was asked to explore the segments (AF), (BD), (EC) and (FB), (DC), (EA) for various triangles and various locations of P.

This picture clearly demonstrates the segments that I am interested in. The first thing I will do is try to determine if the segments CE and EA are a particular ratio to segment AC. I will do this for all sides of any triangle ABC.
Using the above triangle the following ratios are found:
For Segment AC (the sum of segments CE and EA), and CE, and EA;

For segments BC (the sum of BD and DC), BD, and DC;

For segments AB (the sum of AF and FB), AF, and AB;

I had hoped that all of the green segments would have the same ratio to their respective sides of the triangle. Since that did not happen for this triangle, I will not go any further with this investigation.

I will now try a different ratio. What if we sum the green segments and sum the red segments and measure that ratio. There might be something about that ratio that lends itself to investigation.
Starting with the same triangle.

For this triangle the following calculations can be made.

I will now make the same calculations for several triangles, with P remaining fixed, to try to determine if that ratio is always the same.

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