• Given three inline points A,B,C, to construct the harmonic conjugate of C par rapport Ó AB ?
(that is DA/DB = -CA/CB). Hint
As we just saw, drawing parallels requires we already have some !
• Given two pairs of parallel lines, and any line D, to construct a parallel to D through a given point P. Hint
• Given a segment AB and a parallel line,
to construct the midpoint of AB. Hint
And conversedly : given a segment AB and its midpoint, to construct a parallel in P to AB.
Let's go with perpendiculars...
We require of course some right angle somewhere !
• Given a square ABCD and a line d. To construct through P the perpendicular to d.
With this square, it seems we can construct a lot of things, but...
• Show a point which can't be constructed, but can with the compass and straightedge.
• Specifically to construct with straightedge alone the center of the tumbler.
And then :
• Copy a given segment on any line.
• Given two points A and B and a straight line D, to construct the intersection points of D with the circle (undrawn) centered in A with radius AB.
• Given points A,B and C,D to construct the intersection points of the (undrawn) circles centered in A with radius AB and centered in C with radius CD.
We have then proved that all points constructible with compass and straightedge are
also constructible with just the straightedge and the tumbler.
I had right to take the compasses away !