A Course of Plane Geometry for Advanced Students
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1 66 PLANE GEOMETRY [CH.
THEOREM 94. [Newton's Theorem.] If POQ, ROS are two variable chords of a conic, fixed in direction, then the ratio " ^ is constant.
OK . C/o Fig. 82.
Take any other position 0' of O.
Draw the chords F'O'Q', R'O'S' in the given directions.
Let R'S' cut FQ at N; FQ, F'Q' meet at a point at infinity ^.
Since Q, Q', F, F' are finite points, -!i^=i=-^— .
Apply Carnot to the triangle a-NO'.
Th NF.NQ 0.Q' ■ aP O'R' . O'S' ^" NR' .NS" aQ.aF ' O'F' .0'Q'~^'' . O'F' .O'Q _ NF.NQ ..." O'R' .0'S'~NR' .NS" OF. OQ ^ ^P^-_NQ, OR. 0S~ NR' . NS' ' . OF.OQ ^ O'F' .O'Q' OR. OS" O'R' .O'S" Q.E.D.
The standard equations of the ellipse, hyperbola, and parabola can be deduced from Newton's theorem, by making use of the fact (see page 134) that the ellipse and hyperbola have one pair of perpen- dicular conjugate diameters; while the parabola has one diameter which bisects all chords perpendicular to it.
Similarly VI.] CELEBRATED PROPERTIES OF THE CONIC 167 I. THE ELLIPSE AND HYPERBOLA.
What to read after A Course of Plane Geometry for Advanced Students?
You can find similar books in the "Read Also" column, or choose other free books by Durell, Clement V. (Clement Vavasor), B. 1882 to read onlineMoreLess
To quickly assess the difficulty of the text, read a short excerpt:
1 66 PLANE GEOMETRY [CH.
THEOREM 94. [Newton's Theorem.] If POQ, ROS are two variable chords of a conic, fixed in direction, then the ratio " ^ is constant.
OK . C/o Fig. 82.
Take any other position 0' of O.
Draw the chords F'O'Q', R'O'S' in the given directions.
Let R'S' cut FQ at N; FQ, F'Q' meet at a point at infinity ^.
Since Q, Q', F, F' are finite points, -!i^=i=-^— .
Apply Carnot to the triangle a-NO'.
Th NF.NQ 0.Q' ■ aP O'R' . O'S' ^" NR' .NS" aQ.aF ' O'F' .0'Q'~^'' . O'F' .O'Q _ NF.NQ ..." O'R' .0'S'~NR' .NS" OF. OQ ^ ^P^-_NQ, OR. 0S~ NR' . NS' ' . OF.OQ ^ O'F' .O'Q' OR. OS" O'R' .O'S" Q.E.D.
The standard equations of the ellipse, hyperbola, and parabola can be deduced from Newton's theorem, by making use of the fact (see page 134) that the ellipse and hyperbola have one pair of perpen- dicular conjugate diameters; while the parabola has one diameter which bisects all chords perpendicular to it.
Similarly VI.] CELEBRATED PROPERTIES OF THE CONIC 167 I. THE ELLIPSE AND HYPERBOLA.
What to read after A Course of Plane Geometry for Advanced Students?
You can find similar books in the "Read Also" column, or choose other free books by Durell, Clement V. (Clement Vavasor), B. 1882 to read onlineMoreLess
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