Dynamic locus lines
Where do all the points lie from which a given segment appears under a right angle? The following construction will help to answer this (classical) question:
The following drawing provides an additional (dragging) point Z that is bound to a semicircle around A. Varying Z on the semicircle allows to vary the direction of the ray h in the range mentioned above:
- A segment [AB] is given. Now construct a ray h that starts in the point A. The angle between the segment and the ray may vary in the range [-90°.....+90°].
- Raise the perpendicular g from point B on the line h, and mark its foot as C.
When you now record the positions of C, while Z runs through the whole semicircle, you get the locus line of the point C. Obviously, this locus line is the (full) Thales circle over [AB]:
When you drag A or B, not only g, h and C are redrawn, but also the locus line itself is recalculated and updated.
Two questions for further puzzling:
- Why is Z bound to a semicircle and not to a full circle? How does the locus line change when you use a full circle? Construct an appropriate drawing and try it!
- Is it possible to bind Z to a straight line, too? Construct a drawing and try to find the best possible position for this "carrier line"!