## Probability that the circumcenter lies inside the triangle

There was a problem that came up on my coordinate geometry test, which goes like this:

Statement1: Three circles at unequal radii touch each other externally. The point of intersection of the common tangents drawn at the points of contact of the circles is the circum-centre of the triangle formed by joining their centers.

Statement2: The circum-centre is the point of intersection of perpendicular bisectors of the sides of the triangle.

Now, one had to verify the validity of these statements. The second statement was obviously true. As for the first one, one could prove it using general coordinate geometry, but this is a small approach I thought of:

=> It can easily be shown that taken any three arbitrary points, we can draw circles with the points as their respective centers and select radii such that the circles exactly touch each other. Therefore, the set of triangles which can be constructed by this method is actually the set of all real triangles.

A figure for explanation purposes:

Now, I need to find the probability that taken any random triangle, what are the chances that the circum-center of the triangle will lie inside the body of the triangle and not outside it (the area shaded gray, i.e the triangle ABC). Let the probability of this event be A.

Now, I need that the circum-center does not lie in the are shaded dark-gray. Let the probability that the circum-center DOES lie in the area be B.

Then the probability that the circum-center will lie in one of the circles can be given by:

$\mathbf{P} = \mathbf{A} - \mathbf{B}$

Now, if P is not equal to zero, then there exists atleast one triangle such that it's circum-center lies within one of the three circles.

But a tangent never enters a circle as it touches exactly one point on the circle, which means that it will never intersect any other curve inside the circle. Which means that the point of intersection of the tangents cannot be the circum-center of the triangle which lies inside the circle. The statement states for all cases. If there is atleast a single case that it does not satisfy, one can disprove the statement.

Now, it is very obvious that P is not equal to zero, but for mathematical establishment, I need to find out P and/or a mathematical proof that P is not zero.

Thanks a lot.

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