Exploring Point P

 

By: Stephen Bismarck

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Here's a wield relationship that happens when you pick a point inside any triangle.

 

As you can see the product of curtain sides will give you the same result.

Here's the GSP file to check it out for yourself.

Point P

Basically the relationship is AF*BD*EC = DC*AE*FB

To make this easier to prove divide both sides by DC*AE*FB to get

Now to prove that this relationship is true for all points P and any triangle ABC there's only one trick

Look at the areas of the triangles with different bases.

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I have added the altitude HA to show that triangle ABD and triangle ADC have the same altitude.

So the area of triangle ABD = 1/2*HA*BD

and the area of triangle ADC = 1/2* HA*DC

Solving for BD and DC you get

BD = 2*area ABD/HA

DC = 2*area ADC/HA

 

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There are more triangles with those bases

Using the same procedure of finding the area and solving for DB and BC you get

Since the ratio of those two areas are equal then they also equal

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Doing this process for the other triangles you get

 

 

 

 

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By substituting everything in to the original equation

which is equal to 1

 

 

 

 

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