While working in GeoGebra I noticed something odd. I had a triangle with a point inside and the point was connected to each of the vertices. For each vertice I had drawn the circle passing through the vertice and the point, with the connection being the circle's diameter (see picture below).
What I noticed is that the overlapping circles completely covered the triangle. Further experimentation showed this was also the case if the point was outside the triangle (see below).
Still more experimentation appears to show this is the case for any polygon, simple or not:
Is this observation true or did GeoGebra lead me astray? I couldn't immediately find the result via Google.