_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 *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 *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. Circumscribed quadrilateral.* If two pairs of lines in Brianchon’s hexagon coalesce, we have a theorem concerning a quadrilateral circumscribed 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 circumscribed about a conic all meet in a point._ The consequences of this theorem will be deduced later. [Figure 24] FIG. 24 *89. Circumscribed 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 circumscribed about a conic all meet in a point._ *90.* Brianchon’s theorem may also be used to solve the following problems: