For let the vertices be _A_, _B_, _C_, and _D_, and call the vertex _A_ the point 1, 6; _B_, the point 2; _C_, the point 3, 4; and _D_, the point 5 (Fig. 16). Pascal’s theorem then indicates that _L = AB-CD_, _M = AD-BC_, and _N_, which is the intersection of the tangents at _A_ and _C_, are all on a straight line _u_. But if we were to call _A_ the point 2, _B_ the point 6, 1, _C_ the point 5, and _D_ the point 4, 3, then the intersection _P_ of the tangents at _B_ and _D_ are also on this same line _u_. Thus _L_, _M_, _N_, and _P_ are four points on a straight line. The consequences of this theorem are so numerous and important that we shall devote a separate chapter to them. *77. Inscribed triangle.* Finally, three of the vertices of the hexagon may coalesce, giving a triangle inscribed in a conic. Pascal’s theorem then reads as follows (Fig. 17) for this case: _The three tangents at the vertices of a triangle inscribed in a conic meet the opposite sides in three points on a straight line._ [Figure 18] FIG. 18 *78. Degenerate conic.* If we apply Pascal’s theorem to a degenerate conic made up of a pair of straight lines, we get the following theorem (Fig. 18): _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 construct the conic. ("To construct the conic" means here to construct as many other points as may be desired.) 6. Given three points on the conic, and the tangent at two of them, to construct the conic. 7. Given five points, two of which are at infinity in different directions, to construct the conic. (In this, and in the following examples, the student is supposed to be able to draw a line parallel to a given line.) 8. Given four points on a conic (two of which are at infinity and two in the finite part of the plane), together with the tangent at one of the finite points, to construct the conic.