Finding average area of a triangle within circle

[tardy cipher]

I'm trying to find the average area of a triangle formed from 3 random points inside a circle with radius 1.

What I tried doing was defining a length, x, to be the distance from the centre of the circle to the base line of the triangle (the line connecting the first two points).
For all x values from 0 to 1, the average base and height would be calculated, and would be multiplied by the probability of that x value. Halfed, this should equal the average area of the triangle.

I assumed that the probability of a x value would be porportional to the length of the base line, extended to either side of the circle, squared (due to it before for two points). From there, I thought it would be multiplied by a constant such that the probability for all x values summed to 1.

For the average base, I used the average absolute difference of two random points on a line of length 1 (that being 1/4), multiplied by the length of that base line, extended to either side of the circle.

For the average height, I used the average absolute difference between the base line and some height, multiplied by the probability of that height.

The value of this was ~.2 which seems kind of off to me, considering the area of the whole circle is π. Are there any errors in my thinking or any other ways to better approach this problem? Thanks!

[dreamy prism]

Since you're using points to define the triangle it maybe helpful to use the determinant formula of triangle's area rather than the the usual one.