Problem #452

Equilateral triangles 4
Public 01/01/18 9xp Probability 66.7%

Assume a point $D$ inside an equilateral triangle $ABC$ of side length $\sqrt{3}$. Then we can use the three segments $AD$, $BD$ and $CD$ to build a new triangle $PQR$.

When point $D$ is in the center of triangle $ABC$, the area of triangle $PQR$ is $\sqrt{3}/4$, which is the maximum value for all possible positions of $D$; and the circumference of $PQR$ is $3$, which is the minimum value. Moreover, when point $D$ is close to the vertex of triangle $ABC$, the area of triangle $PQR$ is close to zero; and the circumference of $PQR$ is close to its maximum value $2\sqrt{3}$.

Assuming that point $D$ is chosen randomly (with uniform distribution) within triangle $ABC$, find the expected area and circumference of triangle $PQR$. Give your answers rounded to 8 digits after the decimal point.

Answer format: [expectation of area],[expectation of circumference]
(Sample input: 0.43301270,3.46410162)




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