Parallel lines meeting a plane and parallel projection

In Ukrainian Grade 10 (geometry, section 4) parallel lines meeting a plane α are studied. If through points B and C of a segment AB that has the point A in common with α parallel lines are drawn, then CC₁ is a midline of triangle ABB₁ and BB₁ = 2·CC₁.

If AB does not meet α, then AA₁B₁B is a trapezoid and CC₁ (C the midpoint of AB) is its midline: CC₁ = AA₁ + BB₁2.

Parallel projection: through every point of a figure F draw a line parallel to a line l that meets the plane α. The points where these lines meet α form the figure F₁, the parallel projection of F onto α in the direction of l. F₁ is also called the image of F.

Theorem 1. The projection of a line is a line, of a segment a segment. Theorem 2. The projection of two parallel lines is two parallel lines or one line. Theorem 3. The ratio of the projections of segments on one line or on parallel lines equals the ratio of the segments: A₁B₁C₁D₁ = ABCD.