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that oin is in one of the opposite hyperbola S.
Let Gae the oin stom henc GE, GF Fig. 2.are drawn to the socii opposite hyperbolasi; is the exces of the ne traight line bove theotheri equat to the transverse axis a A, thepoliat cis in ne of the opposite hyperbolas. of the two straight lines le GF be the less; and stomuli centre F distance FG describe acircle meeting FE in H; ahe G equat o G and E, b hypothesis, illi equa to the tranSVerse axis Am and because FG, Gnare together greater than E; therei ore FG, GKare together greater than F and Elogether; Consequently FG, or FH, the hal os FG, GK, is greater than FA the hal of A aE; and therei ore the hyperbola, toward the poliat A, talis illi in the circle and Since def. I. 3.)it a b extended beyon an give distancesto the focus F, it necessarii meet the circle.
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No the hyperbola meet the circle in thepsint G sor is not, et it ut it in another po in D, o the fame fide of the axis it the in G and otia DE, DF tbeniec ause thepoin mi in the hyperbola by the first prop. S. the Xcessi DKabovem is equat to the transverse aXi Au buti hypothesis the excessos E ab ove GF is equa to the fame Aa s and FG is equatuo F there fore EG is also equalis ED; hicli is contrar to prop. I. b. I. OfEuclid Theres ore the potnim is in the hyperbola. PROP. III. THEOR,
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Fromuli potnt L,ithout an hyperbola, let Fig. 2. the two traight lines LE, LUbe drawn to the foci the excessis the one bove the ther istes than the transverse axis Aa For sincemis ithout an F within the hyperbola, thestraight line I F necessarii meetsthe hyperbola; leti'meet it in G, and oin EG then EL is Iescthan EG and L; there re the excessis ΕL abovet is tess than the excessis EG and G together above the fame LF that is, hanthe excessi EG above GF that is, Iesa stanthe transverse axisina.
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The Iast par of the proposition, o the Converse of thes no demonstrated is evident. Co R. Hence, i through the vertex A of the tranSverse Xis, a traight line e ram atright angies to that axis, this straight in is wholt without the hyperbola, and consequentdtouches t. For in the traight linem drawn ahe any
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tained by the traight lines be-
I et Aa bo the transverse Xis, C the centre Fig. I. E, F the foci, and B the second axis, hicli, fro the definition of it, is bisecte in thecenire; oi AB and ecatis AB, C are 6 def. 3. equat, the quare of AC, CB aretogether equa to the square of CF that is, to 6 2. I lem. the quare of AC together illi
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the rectangi AFa: tahe way the commoti square of AG and there ill remain the square of CB equat to the rectangle AFa. PROP. V. THEOR.
from ilia potat a stra ight line bedra n to the nearer of the foci; half the transverse Xis is to thodistance e tween that focus and the Centre a the distanc he- tWeen the perpendicular and thecentre, is to the sum os half the tranSverse Xis and the traight line drawn rom heloin to that
fame DCUS. pig. s. 4. I et te a potnt in the hyperbola fronis dra GD perpendicular to the transverse Xiν
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Aas and seo the fame potnt drama traight line F to the nearest secus F; then A, alf the transverse axis is to F the distance belweenthe centre and the focus, s CD, the distancebetween the centre and the perpendicular, is to the sumo half the transverse axis and the straight line drawn to the focus that is, to A together
Dra G to the other secus, and in the aXisa produced place AH equat o GF, and fromth centre G, and distance GF, describe a circle mee ting the axis a A again in Κ, and theatraight
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part AH e taken equa to the distance etween the sint Gand the focus I the quare of the perpendieula GD 4s equalto the exces of the rectangle EHF, contained by the Segments belween the otii H and the foci, above tho rectangles Dacontained by the Segment be- tween the perpendicular and the
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are together equat o twice the rectangle ACH, together illi the quare of AH T. 2. Elem.)that is hec auge CA, CF, CD, CH are proportionali preced. prop. equa tori ic the rectangle FCD, togethe with the square of AH or GF that is, eques t twic the rectangle FCD, togethe with the square of FD DG that is, equat to the sum of the quares of FC, CD, and DG 7. 2. Elem : in there re the two SqUare OfCA, CH are equat to the three squares of CF, CD, DG ut the sum of the two firs is equal 6. 2. Elem. to the squares of A, F, togetherwith the rectangle EHF and the sum of the three Iastris equa 6. 2. Elem. to the quares o CA, CF, G, and the rectangi ADau
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Cular to the transverse axis : the Square of the transverse axis isto the quare of the secon Daris, a me rectangle Contained by the Segments belWeen the perpendicular and the vertices of the trans' Verse Xis is to the square of the Perpendicular.
Fig. 3. 4. et me a Potnt in the hyperbola ist0 Gdrais inperpendicular to the transverse Xis Aa a then the square of Aa i to the quare si Bb, a the rectangi ADa, Containe by the, segments bet een the vertices of the transversu
the foci, place AH roin the ne arest vertecto the focus F, of the transverse axis, equat o GF the lesse of them then, because CH, CD, F, C are proportionals their quare are alio proportionais; ut the square of CH is equaὶ to the square of CF together With the rectangle