*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 circumscribed hexagon, then Brianchon’s theorem may be stated as
follows:

_The three lines joining the three pairs of opposite vertices of a hexagon
circumscribed 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 pass through it. Then _l = (12, 45)_,
_n = (34, 61)_, and the line _m = (23, 56)_ must pass through _(l, n)_.
Then _(23, ln)_ meets 5 in a point of the required sixth line.

[Figure 22]

FIG. 22