. The principles of projective geometry applied to the straight line and conic . irector circles. 139. In Cliaptera conic were given. In this article ccrtaiwith their foci will be i)roved. If LW, VM; LV, WM be any twopairs of parallel tangents to a conic, andA A any other tangent, which meets WLand L V in A and A, then A, A; If, oc ;00, 1^ determine on LW and L V twoprojective ranges, and any other tangentto the conic meets these lines in pairs ofcorresponding points of these ranges.(Art. 93.) By Art. 39 construct points ^S and *Ssuch that corresponding jjoints of theranges subtend a constant
. The principles of projective geometry applied to the straight line and conic . irector circles. 139. In Cliaptera conic were given. In this article ccrtaiwith their foci will be i)roved. If LW, VM; LV, WM be any twopairs of parallel tangents to a conic, andA A any other tangent, which meets WLand L V in A and A, then A, A; If, oc ;00, 1^ determine on LW and L V twoprojective ranges, and any other tangentto the conic meets these lines in pairs ofcorresponding points of these ranges.(Art. 93.) By Art. 39 construct points ^S and *Ssuch that corresponding jjoints of theranges subtend a constant angle a at S,and a constant angle a at »S. Then (Sand *S are the foci of the conic and arethe same points for all pairs of tangentsL W and L V. (Art. 98, Example (8).) From the construction for these points SW. SV= WA . VA = a constant,and the angles A WS and A VS are equal. Let P, Q, P\ Q be the points of con-tact of WL, LV, VM and MW, and C Focal Properties of Conies. XIII, Arts. 9(5 and 97, some tlieorcms connected with the foci ofmetrical properties of conies connected L. the middle point of WV which is also20—2 308 Principles of Projective Geometry the middle point of SS, since SVSW is a parallelogram. Then the angles PSLand LSQ are equal to each other and to the angle ASA. (1) The tangents from any point to a conic are equally inclined to the focaldistances of the point. The angle PWS equals the angle SV(^ (Art. 39) and therefore equals theangle SWi/. (2) The tangent at any point to a conic is equally inclined to the focal distancesof the point. If in (1) the point W is on the conic, the theorem follows at once. (3) //■ a pair of parallel tangents to a conic meet the tangents from a point Win A and B, then WA . WB equals the product of the focal distances of W. If a tangent BB parallel to A A be drawn to meet TFJ/and MV in B and B,then JF=TFZ>. If the tangents L W and L V areparallel, the points F, W, Q and thepoints Q, V, P coincide in two pointsW and V on the co
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