I think I know how geometry got started.
It got started on a clear moonless night when a smart and very ancient Greek looked up and saw points randomly distributed across the sky. He started connecting the dots. Some were lines. Some were triangles. Some were obviously outlines of larger objects.
The Greek started wondering, "Those three points there... what kind of circle can I fit to them? Is the circle unique? Or does more than one circle fit those three points? How can I prove that?"
Then he started sketching star maps in the sand the next day. He sat on a rock and took a stick and used the sharp point to poke star locations here, here, and here. Then he started getting confused.
He had to define a coordinate system, so that the stars could be located according to a fixed coordinate system. That's how coordinate systems came into being.
Angles. The angle of this third star was 45° from the extended line defined by these two nearby stars. And this far away. Thus, the invention of angles, and of polar coordinates.
Cyclic quadrilaterals. Four stars seemed to be shaped so that a circle might fit exactly around them. Was that possible? (Answer: in general, no.) Could you prove that? Let's ask Euclid. He seems to be a smart guy. Or Euxodus. Or Apollonius. They would argue, drink some ouzo, argue some more, call each other shitheads, forget to bathe. Thus, the birth of science conferences.
I lay on my back one cool clear moonless night and stared up at the sky. My eyes saw points of light, but my brain saw lines and triangles. I knew then, at that very moment, that I had to draw a sky map, with a coordinate system, and diagram the stars. My fascination with the geometry of triangles will not be complete until I do this.
And thus, another year-long project is born.
A MAN MUST HAVE PROJECTS Doctor Watson. And you are just such a man. Go to it without hesitation of a backward look. -- the estimable Mister Chu