91Ó°ÊÓ

If \(y / x\) and \(y_{1} / x_{1}\) are two consecutive terms of a Farey series, \(x\) and \(x_{1}\) are coprime.

If a line in the plane of two parallel lines meets one of them, it meets the other also,

Sketch the seven regions into which the lines \(A_{2} A_{3}, A_{3} A_{1}, A_{1} A_{2}\) decompose the plane, marking each according to the signs of the three areal coordinates.

How can a parallelepiped be dissected into six tetrahedra all having the same volume?

If \(y_{0} / x_{0}, y / x_{0}, y_{1} / x_{1}\) are three consecutive terms of a Farey series, $$ \frac{y_{0}+y_{1}}{x_{0}+x_{1}}=\frac{y}{x} $$ (C. Haros, 1802.)

Identify the transformation \((x, y, z) \rightarrow(x, y,-z)\) with the affine reflection that leaves invariant the plane \(z=0\) while interchanging the points \((0,0, \pm 1)\).

Can we always say, of three distinct parallel lines, that one lies between the other two?

If, for a given aftinity, every noninvariant point lies on at least one invariant line, then the aflinity is either a dilatation or a shear or a strain.

In areal coordinates, the midpoint of \(\left(s_{1}, s_{2}, s_{3}\right)\left(t_{1}, f_{2}, t_{3}\right)\) is $$ \left(\frac{s_{1}+t_{1}}{2}, \frac{s_{2}+t_{2}}{2}, \frac{s_{3}+t_{3}}{2}\right) \text {. } $$

Fach of the six edges of a tetrahedron lies on a plane joining this edge to the midpoint of the opposite edge. The six planes so constructed all pass through one point: the centroid of equal masses at the four vertices.