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In a circle with points that fulfull the condition there can not be two points exactly opposite to eachother, since then it would not be possible to get the center stricly inside a triangle made from those two points and a third one.

Now, let anti-point denote the (not allowed) point exactly opposite to a point. It is now easy to see that for the condition to be fulfilled there can not be two anti-points with no point in between, or, equivalently, there can not be two points with no anti-point in between. So, in other words, there has to be every second point and every second anti-point (and of course there has to be at least 3 points, so that you can make a triangle).

In this circle you have every second point and every second antipoint, so the points fulfill the condition. In the following steps we are going to remove the blue points, one by one.

If you take such a situation and remove a point, you will clearly create a situation where this pattern is broken, but you will then always be able to repair the pattern again by removing one more point.

In this circle one of the blue points were removed, and the result is that there are two anti-points with no point in between (or, equivalently, two points with no anti-point in between), so the condition is not fulfilled.	So here we remove the other blue point in order to restore the condition of every second point and every second anti-point.

So, you will be able to leave the situation in good condition only by removing such pairs of points. And you can go on doing this repeatedly as long as there there is a big enough number of points left to fulfill the condition, and the lowest such number is clearly 3, since one must be able to form a trianlge. So you will eventually come to the stage where there are only 3 points left,

Thus there was an odd number of points from the beginning.