Parallel projection and parallelism problems

In parallel projection we choose a plane (the plane of projection) and a line meeting it (the direction). The projection of a point A is the point A₁ where the line through A parallel to the direction meets the plane. The projection of a figure is the figure formed by the projections of all its points; it is like a shadow cast by parallel rays of light. Properties: a straight segment goes to a segment (or a point); parallel segments go to parallel segments (or coincide); the ratio of lengths of segments on one line or on parallel lines is preserved, in particular the midpoint of a segment goes to the midpoint of its projection. But angles and lengths are not preserved: a square can become a parallelogram and a right triangle can become any triangle; a parallelogram never becomes a trapezoid. Medians of a triangle project to medians, whereas altitudes in general do not project to altitudes.