*111.* _In a hyperbola the center is outside the curve_ (§ 101), since the two tangents to the curve at the points where it meets the line at infinity determine by their intersection the center. As previously noted, these two tangents are called the _asymptotes_ of the curve. The ellipse and the parabola have no asymptotes. *112.* _The center of the parabola is at infinity, and therefore all its diameters are parallel,_ for the pole of a tangent line is the point of contact. _The locus of the middle points of a series of parallel chords in a parabola is a diameter, and the direction of the line of centers is the same for all series of parallel chords._ _The center of an ellipse is within the curve._ [Figure 28] FIG. 28 *113. Theorems concerning asymptotes.* We derived as a consequence of the theorem of Brianchon (§ 89) the proposition that if a triangle be circumscribed about a conic, the lines joining the vertices to the points of contact of the opposite sides all meet in a point. Take, now, for two of the tangents the asymptotes of a hyperbola, and let any third tangent cut them in _A_ and _B_ (Fig. 28). If, then, _O_ is the intersection of the asymptotes,—and therefore the center of the curve,— then the triangle _OAB_ is circumscribed about the curve. By the theorem just quoted, the line through _A_ parallel to _OB_, the line through _B_ parallel to _OA_, and the line _OP_ through the point of contact of the tangent _AB_ all meet in a point _C_. But _OACB_ is a parallelogram, and _PA = PB_. Therefore _The asymptotes cut off on each tangent a segment which is bisected by the point of contact._ *114.* If we draw a line _OQ_ parallel to _AB_, then _OP_ and _OQ_ are conjugate diameters, since _OQ_ is parallel to the tangent at the point where _OP_ meets the curve. Then, since _A_, _P_, _B_, and the point at infinity on _AB_ are four harmonic points, we have the theorem _Conjugate diameters of the hyperbola are harmonic conjugates with respect to the asymptotes._ *115.* The chord _A"B"_, parallel to the diameter _OQ_, is bisected at _P’_ by the conjugate diameter _OP_. If the chord _A"B"_ meet the asymptotes in _A’_, _B’_, then _A’_, _P’_, _B’_, and the point at infinity are four harmonic points, and therefore _P’_ is the middle point of _A’B’_. Therefore _A’A" = B’B"_ and we have the theorem _The segments cut off on any chord between the hyperbola and its asymptotes are equal._ *116.* This theorem furnishes a ready means of constructing the hyperbola by points when a point on the curve and the two asymptotes are given. [Figure 29] FIG. 29