Two Planar Constructions Related to a 3D Problem

Elsewhere I discuss a problem of finding a plane that meets an arbitrary pyramid (a shape formed by four rays emanating from the same point and crossing a plane) in a parallelogram. Three solutions were given, two of which depended on constructions in a plane. For the reference sake, I give below the two constructions.

Problem 1

Inscribe a line segment into an angle formed by two rays, equal and parallel to a given one.

Solution

Problem 2

Given three rays emanating from the same point O - one between the "outer" two. Find a triangle with two sides on the outer rays for which the middle ray serves a median.

Solution

Inscribe a line segment into an angle formed by two rays, equal and parallel to a given one.

Solution

Let O be the vertex of the angle. Place one end of the segment on one of the rays - the sides of the angle. Say, let the segment be AB, with A on one of the rays. Through the other end B draw a line parallel to OA. The line will meet the other ray in a point B'. The line through B' parallel to AB will meet OA in A' such that A'B' = AB and A'B'||AB.

Given three rays emanating from the same point O - one between the "outer" two. Find a triangle with two sides on the outer rays for which the middle ray serves a median.

Solution

Pick a point, say M, on the middle ray and turn all three around M 180°:

If O' is the image of O and A and A' are the points of intersection of the rays with the other ray image, the quadrilateral OAO'A' is a parallelogram so that its diagonal AA' is divided into equal parts by point M on the middle ray. AA' is thus a solution to the problem.

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