On the line _a_, joining _A_ and _A'_, take two points, _S_ and _S'_, as centers of pencils perspective to _u_ and _u'_ respectively (Fig. 20). The figure will be much simplified if we take _S_ on _BB'_ and _S'_ on _CC'_.

_SA_ and _S'A'_ are corresponding rays of _S_ and _S'_, and the two 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 pa.s.s through a point._

[Figure 21]

FIG. 21

*85.* To make the notation fit the figure (Fig. 21), make _a=1_, _b = 2_, _u' = 3_, _d = 4_, _u = 5_, _c = 6_; or, interchanging two of the lines, _a = 1_, _c = 2_, _u = 3_, _d = 4_, _u' = 5_, _b = 6_. Thus, by different namings of the lines, it appears that not more than 60 different _Brianchon points_ are possible. If we call 12 and 45 opposite vertices of a circ.u.mscribed hexagon, then Brianchon's theorem may be stated as follows:

_The three lines joining the three pairs of opposite vertices of a hexagon circ.u.mscribed about a conic meet in a point._

*86. Construction of the pencil by Brianchon's theorem.* Brianchon's theorem furnishes a ready method of determining a sixth line of the pencil of rays of the second order when five are given. Thus, select a point in line 1 and suppose that line 6 is to pa.s.s through it. Then _l = (12, 45)_, _n = (34, 61)_, and the line _m = (23, 56)_ must pa.s.s through _(l, n)_.

Then _(23, ln)_ meets 5 in a point of the required sixth line.

[Figure 22]

FIG. 22

*87. Point of contact of a tangent to a conic.* If the line 2 approach as a limiting position the line 1, then the intersection _(1, 2)_ approaches as a limiting position the point of contact of 1 with the conic. This suggests an easy way to construct the point of contact of any tangent with the conic. Thus (Fig. 22), given the lines 1, 2, 3, 4, 5 to construct the point of contact of _1=6_. Draw _l = (12,45)_, _m =(23,56)_; then _(34, lm)_ meets 1 in the required point of contact _T_.

[Figure 23]

FIG. 23

*88. Circ.u.mscribed quadrilateral.* If two pairs of lines in Brianchon's hexagon coalesce, we have a theorem concerning a quadrilateral circ.u.mscribed about a conic. It is easily found to be (Fig. 23)

_The four lines joining the two opposite pairs of vertices and the two opposite points of contact of a quadrilateral circ.u.mscribed about a conic all meet in a point._ The consequences of this theorem will be deduced later.

[Figure 24]

FIG. 24

*89. Circ.u.mscribed triangle.* The hexagon may further degenerate into a triangle, giving the theorem (Fig. 24) _The lines joining the vertices to the points of contact of the opposite sides of a triangle circ.u.mscribed about a conic all meet in a point._

*90.* Brianchon's theorem may also be used to solve the following problems:

_Given four tangents and the point of contact on any one of them, to construct other tangents to a conic. Given three tangents and the points of contact of any two of them, to construct other tangents to a conic._

*91. Harmonic tangents.* We have seen that a variable tangent cuts out on any two fixed tangents projective point-rows. It follows that if four tangents cut a fifth in four harmonic points, they must cut every tangent in four harmonic points. It is possible, therefore, to make the following definition:

_Four tangents to a conic are said to be harmonic when they meet every other tangent in four harmonic points._

*92. Projectivity and perspectivity.* This definition suggests the possibility of defining a projective correspondence between the elements of a pencil of rays of the second order and the elements of any form heretofore discussed. In particular, the points on a tangent are said to be _perspectively related_ to the tangents of a conic when each point lies on the tangent which corresponds to it. These notions are of importance in the higher developments of the subject.

[Figure 25]

FIG. 25

*93.* Brianchon's theorem may also be applied to a degenerate conic made up of two points and the lines through them. Thus(Fig. 25),

_If __a__, __b__, __c__ are three lines through a point __S__, and __a'__, __b'__, __c'__ are three lines through another point __S'__, then the lines __l = (ab', a'b)__, __m = (bc', b'c)__, and __n = (ca', c'a)__ all meet in a point._

*94. Law of duality.* The observant student will not have failed to note the remarkable similarity between the theorems of this chapter and those of the preceding. He will have noted that points have replaced lines and lines have replaced points; that points on a curve have been replaced by tangents to a curve; that pencils have been replaced by point-rows, and that a conic considered as made up of a succession of points has been replaced by a conic considered as generated by a moving tangent line. The theory upon which this wonderful _law of duality_ is based will be developed in the next chapter.

PROBLEMS