In projective geometry and curved lines

Question

For example, Fano's Geometry exemplifies a triangle with a circle within it. How is this possible? Are lines not defined to be straight? Is this geometry projective geometry(or is projective geometry a different set of axioms)?

Answer 1

The axioms which define a projective plane only speak about points, lines and incidences. They say nothing about straightness, they say nothing about an embedding into $\mathbb R^2$. The common representation as a triangle with an inscribed circle is just a combinatoric illustration. The circle in the center is just a device to denote that the three points it connects have a line in common. But the points are not really at the positions where that picture shows them. They cannot really be correctly shown in the plane at all. The right way to see them is algebraically, in $\mathbb Z_2\mathbb P^2$, and algebraically that “circular” line is no different from the other six lines.