Question

Suppose that in a Hilbert plane, lines m and n are parallel. Let P be a point on m, and Q the point on n so that line P Q is perpendicular to n. Is there another point R on m so that the perpendicular line RS to n through R (where S is incident with n) so that segment P Q is congruent to RS? Under what conditions? (Note that you need to provide some further hypotheses to make a clear statement about the existence or non-existence of such additional points.)

Answer #1

I have attached an Image herewith. Hope you will be able to understand the solution from it.

Let P be the plane given by the equation 2x + y − 3z = 2. The
point Q(1, 2, 3) is not on the plane P, the point R is on the plane
P, and the line L1 through Q and R is orthogonal to the plane P.
Let W be another point (1, 1, 3). Find parametric equations of the
line L2 that passes through points W and R.

Consider a cicle with AB as diameter and P another point on the
circle. Let M be the foot of the perpendicular from P to AB. Draw
the circles which have AM and M B respectively as diameters, which
meet AP at Q are BP are R. Prove that QR is tangent to both
circles.
Hint: As well as the line QR, draw in the line segments M Q and
M R.

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