A first course in projective geometry . Fig. 336. Conversely, if four lines through a point form a liarmonicpencil, conjugate rays of which are tangents from the point,the other two are conjugate lines with respect to the circle. This follows easily : the proof is left as an exercise. HARMONIC PROPERTIES OF THE INSCRIBED QUAD-RANGLE AND CIRCUMSCRIBED QUADRILATERAL. § 11. Prop. If a quadrangle be inscribed in a circle, thediagonal points determine a triangle self-conjugate withrespect to the circle. Let ABCD (Fig. 34a) be the quadrangle, E, F, G its diagonalpoints. Let EG meet FA, FB in H, K. 7
A first course in projective geometry . Fig. 336. Conversely, if four lines through a point form a liarmonicpencil, conjugate rays of which are tangents from the point,the other two are conjugate lines with respect to the circle. This follows easily : the proof is left as an exercise. HARMONIC PROPERTIES OF THE INSCRIBED QUAD-RANGLE AND CIRCUMSCRIBED QUADRILATERAL. § 11. Prop. If a quadrangle be inscribed in a circle, thediagonal points determine a triangle self-conjugate withrespect to the circle. Let ABCD (Fig. 34a) be the quadrangle, E, F, G its diagonalpoints. Let EG meet FA, FB in H, K. 70 PROJECTIVE GEOMETRY Then (Chap. IV. § 8) FDHA and FCKB are harmonic ranges;.*. (§ 9, above) H and K are points on the polar of Fig. 34a. This polar is therefore EG. Similarly FG is the polar of E ;.*. FE is the polar of G, and the triangle EFG is self-conjugate. § 12. Prop. If a quadrilateral be described about acircle, its diagonal triangle is self-conjugate with respect tothe circle. Let abed (Fig. 346) be the quadrilateral, and efg its diagonaltriangle. Let h, k be the joins of {eg) to (fa) and (fh) respectively. Then (Chap. IV. § 8) a, /, r/, h form a harmonic pencil. So also do h, /, c, k. .*. (§ 9) li and k are each conjugate lines to/. .*. they meet at the pole of/, {eg) is the pole of /. HARMONIC PROPERTIES Similarly the point {gf) is the pole of e. .. the triangle efg is self-con jugate for the circle. 71
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