First, we create a line through A that is parallel to BC.
This allows us to look at several similar triangles and manipulate the properties of those similar triangles.
The first set of similar triangles is: triangle AHF and Triangle BCF.
Corresponding sides of similar triangles have proportional sides. This allows us to look at the extended proportion:
The second set of similar triangles is: triangle AEG and triangle CEB.
Here we look at the extended proportion:
Triangles AGP and DBP are also similar.
The proportion of the corresponding sides is:
And the last set of similar triangles that we need to look at is
triangle CDP and triangleHAP.
Here the proportion relating the sides is:
Using the last two proportions we see that:
Using the Transitive Property, we see that
Using the cross product property of proportions we can rewrite the proportion as:
So we have said:
Multiplying we see:
Which simplifies to:
This holds even when P is outside of the triangle, except when P is on one of the lines that include a vertex and are parallel to the opposite side of the triangle.
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