Fox 240

http://www.8foxes.com/

Now this asks little more than geometry.

Note that the random process described in the question is discrete.

Ladybugs "appear" in next vertices at the same time.

Fox 229

http://www.8foxes.com/

Fox 217

Have beautiful days with your families

and the loved ones!

See you guys next year!

http://www.8foxes.com/

Yourself should show mercy on yourself.
Break your daily routine,
Do something out of the box.
While enjoying the time...
NEVER QUIT THINKIN' !

Fox 201

http://www.8foxes.com/

Fox 199

http://www.8foxes.com/

Fox 198

http://www.8foxes.com/

This is the beginning of a class of foxes with random processes.

The above process is a special form of a random walk on the plane.
We believe that this process is new! If you see something similar in literature, please let us know.

"Uniformly selected" means that every direction has an equal chance of being selected.
In other words: direction ~ Uniform[0, 360].
Point O is the origin. Generation of point A is irrelevant since OA does not change whether there will be a triangle or not at the end of the process.
Also, it is not important but assume that the random directions are generated clock wise.
The answer has not been confirmed yet, let us know if you find something very different.