pencils are therefore in perspective position. It is not difficult to see
that the axis of perspectivity _m_ is the line joining _B’_ and _C_. Given
any point _D_ on _u_, to find the corresponding point _D’_ on _u’_ we
proceed as follows: Join _D_ to _S_ and note where the joining line meets
_m_. Join this point to _S’_. This last line meets _u’_ in the desired
point _D’_.

We have now in this figure six lines of the system, _a_, _b_, _c_, _d_,
_u_, and _u’_. Fix now the position of _u_, _u’_, _b_, _c_, and _d_, and
take four lines of the system, _a__1_, _a__2_, _a__3_, _a__4_, which meet
_b_ in four harmonic points. These points project to _D_, giving four
harmonic points on _m_. These again project to _D’_, giving four harmonic
points on _c_. It is thus clear that the rays _a__1_, _a__2_, _a__3_,
_a__4_ cut out two projective point-rows on any two lines of the system.
Thus _u_ and _u’_ are not special rays, and any two rays of the system
will serve as the point-rows to generate the system of lines.




*84. Brianchon’s theorem.* From the figure also appears a fundamental
theorem due to Brianchon:

_If __1__, __2__, __3__, __4__, __5__, __6__ are any six rays of a pencil
of the second order, then the lines __l = (12, 45)__, __m = (23, 56)__,
__n = (34, 61)__ all pass through a point._

[Figure 21]

FIG. 21