TITLE: Probability that one of a set of four points lies within the triangle formed by the other three
QUESTION [9 upvotes]: Given four points, each randomly chosen with a uniform probability distribution in the interior of a (WLOG unit) circle, what is the probability that (any) one of the points lies within the triangle formed by the other three. This is (meant to be) equivalent to asking what is the probability, given these four random points, that one can form two "layers" of triangles, with for example ABC covered by (though sharing a side with) ABD because C lies in the interior of ABD.
Ideally, the method of solution would be generalizable to numbers of points greater than four; in other words, given (3+n) points (for n>1), each randomly chosen with a uniform probability within the interior of a circle, what is the probability that one can form n+1 layers. (I'm having trouble stating the generalized problem precisely; I hope it's clear.)
Edit to add: In case anyone is curious about whence the problem comes: it was inspired by Ingress, an enhanced reality game run by Google wherein players go to and link predesignated geographic points to form "fields", which are simply completed triangles but which are limited by the rule that links can never cross each other. In the game it is often advantageous to form layers of fields, the simplest case of which, for four points, is what I describe in the first paragraph and the most efficient general case (n-2 overlapping fields out of n points) I allude to in the second. I've seen and heard many comments by players about how often such overlapping fields can be formed, but I've never seen anything approaching a definitive answer to it.
Note that this means the question I really wanted to ask is the same one but for points on the surface of a sphere, not a portion of a flat plane, but almost all fields in the game are quite small relative to the Earth (though not all; on rare occasions, fields with edge lengths of thousands of kilometers are formed; the great majority are tens or hundreds of meters) so the answer to the question I posed will be a very good approximation.

REPLY [3 votes]: As the estimate answer is a bit long winded, I've posted this mathematical answer separately.
Given a random triangle in a circle, the chance of a random point landing in the triangle is the area of a triangle divided by the area of a circle.
So what we need is the average area of a triangle in a circle. The average area of a triangle for a unit disk (r=1) can be determined with the following formula as per Worlfram Disk Triangle Picking 
$$A_t = \frac{35}{48 \pi} \approx 0.232100... $$
As we all know the area of a circle is
$$A_c = \pi r^2 $$
Because r=1 for a unit disk it is simplified to
$$A_c = \pi $$
So the chance of the fourth random dot landing in a random triangle is
$$p_4 = \frac{A_t}{A_c} = \frac{35}{48 \pi^2} \approx 0.07388003$$
However with four points we are forming four triangles, each point will have the same chance to land in the triangle formed by the three other points, so the probability is.
$$p = 4 \frac{A_t}{A_c} = 4 \frac{35}{48 \pi^2} =  \frac{35}{12π^2} \approx 0.295520119$$
The above formula is only valid for

A unit disk (r=1). I've not yet come across any proofs for the average area of a triangle in a circle with radius r.  Does it increase at the same rate as the area of the circle at $r^2$?*  
Four random points only. The extended question asking for more points is not yet covered.

EDIT: Probability remains the same for all r > 0
Now I've looked at more than 4 points as well and generated sets of random points (100000 times) and tallied up how many points landed in a triangles. Some interesting results which I've not worked out the math for yet.
╔═════════════╦═════════╦═════════╦═════════╦═════════╦═════════╗
║      Points ║    4    ║    5    ║    6    ║    7    ║    8    ║
╠═════════════╬═════════╬═════════╬═════════╬═════════╬═════════╣
║ Triangles   ║    4    ║    10   ║    20   ║    35   ║    56   ║
║ Any         ║ 0.29845 ║ 0.64469 ║ 0.86461 ║ 0.96053 ║ 0.99096 ║
║ 0           ║ 0.70155 ║ 0.35531 ║ 0.13539 ║ 0.03947 ║ 0.00904 ║
║ 1           ║ 0.29845 ║ 0       ║ 0       ║ 0       ║ 0       ║
║ 2           ║ 0       ║ 0.54789 ║ 0       ║ 0       ║ 0       ║
║ 3           ║ 0       ║ 0       ║ 0.24551 ║ 0       ║ 0       ║
║ 4           ║ 0       ║ 0.0968  ║ 0.20193 ║ 0.08295 ║ 0       ║
║ 5           ║ 0       ║ 0       ║ 0.04365 ║ 0       ║ 0.02192 ║
║ 6           ║ 0       ║ 0       ║ 0.12658 ║ 0.12148 ║ 0       ║
║ 7           ║ 0       ║ 0       ║ 0.14877 ║ 0       ║ 0       ║
║ 8           ║ 0       ║ 0       ║ 0.06697 ║ 0.12974 ║ 0.02969 ║
║ 9           ║ 0       ║ 0       ║ 0.00447 ║ 0       ║ 0.01231 ║
║ 10          ║ 0       ║ 0       ║ 0.01476 ║ 0.16427 ║ 0.01963 ║
║ 11          ║ 0       ║ 0       ║ 0.0096  ║ 0       ║ 0.01344 ║
║ 12          ║ 0       ║ 0       ║ 0.00237 ║ 0.19293 ║ 0.01909 ║
║ 13          ║ 0       ║ 0       ║ 0       ║ 0       ║ 0.06144 ║
║ 14          ║ 0       ║ 0       ║ 0       ║ 0.11626 ║ 0.02674 ║
║ 15          ║ 0       ║ 0       ║ 0       ║ 0       ║ 0.00612 ║
║ 16          ║ 0       ║ 0       ║ 0       ║ 0.08477 ║ 0.05515 ║
║ 17          ║ 0       ║ 0       ║ 0       ║ 0       ║ 0.05771 ║
║ 18          ║ 0       ║ 0       ║ 0       ║ 0.04646 ║ 0.05092 ║
║ 19          ║ 0       ║ 0       ║ 0       ║ 0       ║ 0.04256 ║
║ 20          ║ 0       ║ 0       ║ 0       ║ 0.01522 ║ 0.04368 ║
║ 21          ║ 0       ║ 0       ║ 0       ║ 0       ║ 0.07326 ║
║ 22          ║ 0       ║ 0       ║ 0       ║ 0.00392 ║ 0.05944 ║
║ 23          ║ 0       ║ 0       ║ 0       ║ 0       ║ 0.03105 ║
║ 24          ║ 0       ║ 0       ║ 0       ║ 0.00192 ║ 0.0459  ║
║ 25          ║ 0       ║ 0       ║ 0       ║ 0       ║ 0.06104 ║
║ 26          ║ 0       ║ 0       ║ 0       ║ 0.00061 ║ 0.04756 ║
║ 27          ║ 0       ║ 0       ║ 0       ║ 0       ║ 0.02972 ║
║ 28          ║ 0       ║ 0       ║ 0       ║ 0       ║ 0.03464 ║
║ 29          ║ 0       ║ 0       ║ 0       ║ 0       ║ 0.03928 ║
║ 30          ║ 0       ║ 0       ║ 0       ║ 0       ║ 0.02749 ║
║ 31          ║ 0       ║ 0       ║ 0       ║ 0       ║ 0.01559 ║
║ 32          ║ 0       ║ 0       ║ 0       ║ 0       ║ 0.01359 ║
║ 33          ║ 0       ║ 0       ║ 0       ║ 0       ║ 0.01606 ║
║ 34          ║ 0       ║ 0       ║ 0       ║ 0       ║ 0.01174 ║
║ 35          ║ 0       ║ 0       ║ 0       ║ 0       ║ 0.00531 ║
║ 36          ║ 0       ║ 0       ║ 0       ║ 0       ║ 0.00451 ║
║ 37          ║ 0       ║ 0       ║ 0       ║ 0       ║ 0.00515 ║
║ 38          ║ 0       ║ 0       ║ 0       ║ 0       ║ 0.00333 ║
║ 39          ║ 0       ║ 0       ║ 0       ║ 0       ║ 0.00132 ║
║ 40          ║ 0       ║ 0       ║ 0       ║ 0       ║ 0.00119 ║
║ 41          ║ 0       ║ 0       ║ 0       ║ 0       ║ 0.00121 ║
║ 42          ║ 0       ║ 0       ║ 0       ║ 0       ║ 0.0009  ║
║ 43          ║ 0       ║ 0       ║ 0       ║ 0       ║ 0.00036 ║
║ 44          ║ 0       ║ 0       ║ 0       ║ 0       ║ 0.00014 ║
║ 45          ║ 0       ║ 0       ║ 0       ║ 0       ║ 0.00025 ║
║ 46          ║ 0       ║ 0       ║ 0       ║ 0       ║ 0.00017 ║
║ 47          ║ 0       ║ 0       ║ 0       ║ 0       ║ 0.00009 ║
║ 48          ║ 0       ║ 0       ║ 0       ║ 0       ║ 0.0001  ║
║ 49          ║ 0       ║ 0       ║ 0       ║ 0       ║ 0.0001  ║
║ 50          ║ 0       ║ 0       ║ 0       ║ 0       ║ 0.00006 ║
║ 51          ║ 0       ║ 0       ║ 0       ║ 0       ║ 0.00001 ║
║ 52          ║ 0       ║ 0       ║ 0       ║ 0       ║ 0       ║
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