Unizor - Geometry3D - Lines and Planes - Problems 7

Unizor - Creative Minds through Art of Mathematics - Math4Teens

Below are a few construction problems.
Recall certain principles of construction of planes in three-dimensional space that we have agreed upon:
The plane is considered as completely defined and uniquely constructed if
(a) there are three points not on the same line that are known to belong to a plane;
(b) there is a line and a point outside of this line that are known to belong to a plane;
(c) there are two intersecting lines that are known to belong to a plane;
(d) there are two parallel lines that are known to belong to a plane.
If two non-parallel planes are constructed, we assume that their intersection is fully defined and constructed.
If a plane and a non-parallel to it line are constructed, we assume that their intersection is fully defined and constructed.

In the previous lectures we discussed some elementary construction problems like constructing a line or a plane in three-dimensional space parallel or perpendicular to a given line or plane passing through a given point or line. We recommend to refresh that material prior to addressing the problems below.

Problem 1
Construct a plane that passes through a given point M and is parallel to two given skew lines a and b.
?γ: γ∋M, γ ∥ a, γ ∥ b

Problem 2
Construct a line that passes through a given point M, intersects a given line a and is parallel to a given plane γ.
?b: b∋M, b∩a≠∅, b ∥ γ

Problem 3
Construct a line that intersects two given lines a and b and is parallel to the third line c.
?h: h∩a≠∅; h∩b≠∅; h ∥ c

Problem 4
Construct a line that passes through a given point H and is perpendicular (not necessarily intersecting) to two given lines a and b.
?h: h∋H; h⊥a; h⊥b

Problem 5
Construct a plane that intersects a given plane γ at a given angle ∠φ and contains a given line a that is parallel to plane γ.
a∩γ=∅; a ∥ γ;
?δ: δ∋a; ∠(δ,γ)=∠φ

Problem 6
Given a plane γ and two points A and B outside of this plane and lying in the same half-space (considering plane γ as a border between two half-spaces).
Find a point M on plane γ such that a sum of lengths of segments AM+MB is minimal.
A∉γ; B∉γ; AB∩γ=∅;
?M: AM+MB = min.