and if it touch the hyperbola, it is bisected in the point of contact.

Let there be an hyperbola (fig. 15. n. 1.) the asymptotes of which are AB, AC, and let a straight line BC, terminated by the asymptotes, meet it in the point D, and be bisected in D; the straight line BC touches the hyperbola.

Through D draw DE parallel to the one asymptote AC, and meeting the other in E; and in BC take any point G, through which draw GH parallel to DE; GH will meet the hyperbola (18.3) in some point F: then, because BD, DC are equal, BE, EA are also equal; and, because of the equiangular triangles, BE is to ED, as BH to HG; therefore (1.6. Elem.) the rectangle BEA is to the rectangle DEA, as the rectangle BHA to GHA: but the rectangle BEA is (5. 2. Elem.) greater than BHA; therefore the rectangle DEA is also greater (14. 5. Elem.) than the rectangle GHA; that is, because F is in the hyperbola, the rectangle FHA is greater than the rectangle GHA; and therefore FH is greater than HG: therefore the point G is without the hyperbola; and therefore the straight line BC touches the hyperbola in the point D.

OTHERWISE.

If a straight line LM (fig. 15. n. 1.) terminated by the asymptotes, is bisected by the hyperbola in the point D, it touches the hyperbola in this point.

It is plain that the straight line LD passes not within the hyperbola; for if it passed within the hyperbola it would necessarily meet it again in another point, because the points L, M are without the hyperbola: but it is impossible for it to meet the hyperbola in any other point but D. For, if possible, let it meet it likewise in N; therefore NM is (14. 3.) equal to DL, that is, according to the hypothesis, to DM: which is absurd. Therefore LM falls not within the hyperbola, nor meets it any where but in the point D; and therefore LM touches it in D.

On the contrary: if the straight line LM, terminated by the asymptotes, touch the hyperbola in D, it is bisected in the point of contact.

For if LD, DM are unequal, from DM the greater take away MN equal to LD the less; therefore the point N is (19. 3.) in the hyperbola; and therefore, contrary to the hypothesis, LM cuts the hyperbola.

Cor. 1. Hence through the same point of an hyperbola (fig. 15. n. 1.) only one straight line can be drawn touching the hyperbola.

Let D be a point in the hyperbola, and through that point to the asymptote AB draw a straight line DE parallel to the other; and take EB equal to EA, and having joined BD, let it meet the asymptote AC in C: then, since BE, EA are equal, BD, DC are also equal; BC, therefore, touches the hyperbola in D. And no other straight line can touch it in the same point D: for, if possible, let LDM also touch it; then, since BE, EA are equal, therefore LE, EA are unequal; and

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consequently LD, DM are likewise unequal: therefore LM does not touch the hyperbola.

COR. 2. Hence is manifest, the manner by which, if the asymptotes AB, AC of an hyperbola be given in position, a straight line BC can be drawn, which shall touch the hyperbola in a given point D.

COR. 3. If through the vertices (fig. 15. n. 2.) of a transverse diameter two straight lines be drawn touching the hyperbolas, they are parallel to each other. Let AC, BC be the asymptotes, and let AOB, QPR touch the hyperbolas in the vertices of the transverse diameter OCP; the tangents AB, QR are parallel. Draw to either asymptote AC the straight lines OS, PT parallel to the other, and the triangles SCO, TCP are equiangular; by the proposition, AO, OB are equal, and because of the parallels, AS, SC are also equal; and, in like manner, QT, TC are equal; and CO is to CP, as CS to CT, and consequently, as CA to CQ; therefore the triangles OCA, PCQ are equiangular; and therefore OA, PQ are parallel.

Cor. 4. And if a straight line be drawn parallel to a tangent, and meeting the hyperbola; the square of the segment of the tangent between the point of contact and either of the asymptotes, is equal to the rectangle contained by the segments of the parallel, between either point of concourse with the hyperbola, and the asymptotes. For this rectangle is equal to the rectangle contained by the segments of the tangent (15. 3.) between the point of contact and the asymptotes, that is, equal

to the square of its segment between the point of contact and either of the asymptotes.

PROP. XXIV. PROВ.

The asymptotes AB, AC (fig. 15. n. 1.) of an hyperbola, and a point F in the same, being given in position; to draw a straight line which shall touch the hyperbola, and be parallel to a straight line KO, which is given in position, and cuts both the asymptotes of the hyperbola, or opposite hyperbolas.

Suppose the problem solved; and let BC be parallel to KO, and touch the hyperbola in D; and having joined AD, let AD meet KO in P; draw FRQ parallel to AD, and meeting the asymptotes in Q, R; and since the straight line BC touches the hyperbola in D, therefore BD is equal to DC (23. 3.) and consequently KP is equal to PO; and KO is given in position and magnitude; therefore KP and the point Pare given; but the point A is given; therefore the straight line PAD is given in position. Now the square of AD is equal to (1. cor. 15. 3.) the rectangle QFR; and since FRQ is given in position (28. dat.) and that AB, AC are likewise given in position; therefore FQ, FR are (25. 26. dat.) given; and therefore the rectangle QFR is given; consequently the square of AD is given; and therefore AD is given in magnitude: but the point A is given in position; therefore the point D is also given (27. dat.) and (28. dat.) therefore the straight line BDC is given in position.

The composition is thus: let KO be bisected in P; and having joined AP, draw through the point Fa straight line FRQ parallel to AP, and meeting the asymptotes in the points R, Q; in AP produced, and in either direction from the centre, take AD a mean proportional between FQ, FR; and through D draw BDC parallel to KO; then BC will touch the hyperbola in D. For since the square of AD is equal to the rectangle QFR, the point Dis (cor. 17. 3.) in the hyperbola; and since KO, BC are parallel, and that KO is bisected in P by the straight line PAD, therefore BC is bisected in D; and consequently touches the hyperbola in the same point D (23. 3.)

PROP. XXV. THEOR,

If two straight lines touch an hyperbola, or opposite hyperbolas, and cut the asymptotes; the rectangle contained by the abscissas of the asymptotes between the centre and the one straight line, is equal to the rectangle contained by the abscissas between the centre and the other straight line.

Let there be an hyperbola (fig. 16.) with AB, AD for its asymptotes, let a straight line BD touch it in C, and let another straight line GE touch the same, or the opposite hyperbola, in F; the rectangles BAD, EAG will be equal.

From the points C, F draw CH, CK, and FL, FM parallel to the asymptotes; then because BCD touches the hyperbola, BC is equal to CD (23. 3.) and conse

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