Prove area equation

[Math Tutor]

If M is a medicenter of a triangle ABC and m is a line through M that intersects AC and BC at points P and N prove that
area(APN) + area(BPN) = area(CPN)

[Guest]

1). Use Cartesian coordinate system and matrix determinant of an triangle and prove the
equality. (this solution requires the vertices of the points)

2). Use Menelaus' theorem and some easy tricks of the triangle in order to solve it in an analytical
manner.


YS.

[Math Tutor]

It is a high school problem. You cannot use Cartesian coordinate system and matrix determinant of an triangle.

Can you solve the problem without Menelaus' theorem?

[Guest]

When i was 12 years old i learned at school Menelaus` theorem. I see no reason to spend my time providing
with more solutions when i`ve already had 2 solutions at hand. These solutions were found just at a mere sight
of the problem.

YS.

[Guest]

I did learn *Cartesian coordinate system and matrix determinant of a triangle area* when i was
15 years old. By the way, i live in Romania and our school is pretty good.

YS.

[Math Tutor]

The problem must be solved without Menelaus' theorem.

[Guest]

At this moment i have at hand the most elementary way to solve that problem. Hint: the areas of the nearby triangles and
the opposite ones, those formed by medians and that line that passes through the mediacenter, are very interesting. Area formula of the triangle S= sin(alfa) *b*c/2 seems also very interesting. Did i say enough? I think so.

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