Mathematical game: what is Euclid’s postulate?

In The elements ofEuclid (III^and century before our era), the postulate comes after the definitions of points, lines, parallelism (two straight lines are parallel if they do not intersect), angles, etc.

The sum of the angles of a triangle

The postulate is related to the fact that the sum of the angles of a triangle is equal to 180°. The demonstration of this property opens the way to non-Euclidean geometriesthat is, those that deny the postulate.

Consider a triangle ABC and extend side AB to BE. Then, from point B, take the line BD so that the angle CBD is equal to angle ACB (both in dark blue). Similarly, let’s draw the line BD’ so that the angle EBD’ is equal to the angle BAC (in light blue). Lines BD and BD’ are parallel to line AC. If the postulate is true, they are identical since, from a point, one can only draw one parallel to a given straight line. The three angles of the triangle ABC (the third in red) thus refer to B to form a flat angle, that is to say 180°. We have thus demonstrated that the sum of the angles of a triangle is equal to 180°… if the Euclid’s postulate is right. When we draw the previous figure on a sheet of paper, the lines BD and BD’ coincide.

By bending the plan…

Let’s cut the paper along the half-line BD and move BD’ on BD, the sheet bends. It becomes like a mountain peak and the sum of the angles of the triangle, greater than 180°. On the contrary, by removing BD’ from BD, the sheet bends in the other direction. It becomes like a mountain pass and the sum of the angles of the triangle, less than 180°. The first is the spherical geometry or Riemann, the second, hyperbolic geometry or Lobachevsky (1792 – 1856).