_If three points, __A__, __B__, __C__, are chosen on one line, and three
points, __A’__, __B’__, __C’__, are chosen on another, then the three
points __L = AB’-A’B__, __M = BC’-B’C__, __N = CA’-C’A__ are all on a
straight line._




PROBLEMS


1. In Fig. 12, select different lines _u_ and trace the locus of the
center of perspectivity _M_ of the lines _u_ and _u’_.

2. Given four points, _A_, _B_, _C_, _D_, in the plane, construct a fifth
point _P_ such that the lines _PA_, _PB_, _PC_, _PD_ shall be four
harmonic lines.

_Suggestion._ Draw a line _a_ through the point _A_ such that the four
lines _a_, _AB_, _AC_, _AD_ are harmonic. Construct now a conic through
_A_, _B_, _C_, and _D_ having _a_ for a tangent at _A_.

3. Where are all the points _P_, as determined in the preceding question,
to be found?

4. Select any five points in the plane and draw the tangent to the conic
through them at each of the five points.

5. Given four points on the conic, and the tangent at one of them, to