# Expected area of a triangle where 1 point is within another triangle

Expected area of a triangle where 1 point is within another triangle
Suppose we have triangle ABC with area k with a point P chosen inside ABC. What is the expected area of triangle PBC?
I'm pretty sure if we let P be the centroid we get k/3. Also, how would you solve this question for an n-sided polygon?
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Marlene Welch
Step 1
Let the vertices of the triangle be a, b, c. Then the map

maps the arbitrary triangle from
$\left(x,y\right)=\left({a}_{1},{a}_{2}\right),\left({b}_{1},{b}_{2}\right),\left({c}_{1},{c}_{2}\right)$ bijectively onto a right triangle with $\left(u,v\right)=\left(0,0\right),\left(1,0\right),\left(0,1\right)$ (make a sketch for visualization). Therefore the transformation formula for multiple integrals can be used, and we obtain

The Jacobian determinant is a constant: From (1) we obtain

Step 2
Therefore we can write

The area of an arbitrary triangle a,b,p is $f=1/2\cdot \text{base}\cdot \text{height}$. In u,v coordinates the base is 1 and the height is v. For normalization the integral has to be divided by the total area that is 1/2. The expected area of a random triangle a,b,p is therefore

As the area of the triangle a, b, c is
${A}_{\mathbf{a},\mathbf{b},\mathbf{c}}=\frac{1}{2}|\left({b}_{1}-{a}_{1}\right)\left({c}_{2}-{a}_{2}\right)-\left({c}_{1}-{a}_{1}\right)\left({b}_{2}-{a}_{2}\right)|$
the expected area of the random triangle is 1/3 of the area of the original triangle, independently which side is used as a base.
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