. The principles of projective geometry applied to the straight line and conic . 6. P, Q are given points on the base AB of a triangle ABC. Find points XYon the sides AC, BC or their productions, such that ifPX meet BC in U and QT meet AC in V, AT shallbe parallel, and ^Mperpendicular, to AB. Take any point X on AC. Draw XY parallel toAB to meet CB in Y. Join YQ meeting AC in V(7 perpendicular to AB to meet CB in UP to meet J C in A. Then the ranges X andA are jjrojective. Take X at J, Y is then at Zf, T^ is at A, U is at U\A is at A. Take X at C, F is then at C, V is at C,t/isat
. The principles of projective geometry applied to the straight line and conic . 6. P, Q are given points on the base AB of a triangle ABC. Find points XYon the sides AC, BC or their productions, such that ifPX meet BC in U and QT meet AC in V, AT shallbe parallel, and ^Mperpendicular, to AB. Take any point X on AC. Draw XY parallel toAB to meet CB in Y. Join YQ meeting AC in V(7 perpendicular to AB to meet CB in UP to meet J C in A. Then the ranges X andA are jjrojective. Take X at J, Y is then at Zf, T^ is at A, U is at U\A is at A. Take X at C, F is then at C, V is at C,t/isatC, ATis at C. Therefore C is a self-corresponding point of thesuperposed ranges. The required point Z the other self-corresponding point isfound from the relation {ACZX) = {ACZX),. whence AZ AX cz ex AZAZ A^ A^ CZ • CX JACX AXCX CHAPTER X PROJECTIVE FORMS IN RELATION TO THE CIRCLE :—ANHARMONICPROPERTY. POLE AND POLAR. CIRCLES IN SELF-PERSPECTIVE 72. The Circle. Introductory. A circle is usually defined as a curve such that thedistance of every point on it from a fixed point is constant. It im-mediately follows (see Art. 73) that it is the locus of the points ofintersection of corresponding rays of two directly equal pencils. InArt. 92 a conic is defined as the locus of the points of intersection ofpairs of corresponding rays of two projective pencils. Hence a circle isa particular case of a conic. Also since equal pencils are projected intoprojective pencils, the projection of a circle is a conic and since theprocess of projection is reversible a conic may be projected into a would be possible to consider the projective properties of a conic andhence infer those of a circle. One method, however, of approaching thesubject is to deduce by projection
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