. The principles of projective geometry applied to the straight line and conic . s. (c) corresponding chords ABand AB intersect on s. (d) tangents from S tSV=VN. Then the line parallel to s throughV is the vanishing line. The two columns above simply give a restatement of the sametheorem looked at from two slightly different points of view and forma good instance of the Principle of Duality. 80. Complete Inscribed Quadrangle and Complete Circum-scribed Quadrilateral of a Circle. Any four points A, B, C, B ona circle determine (1) three points S, T, U, asthe points of intersection of theirconne
. The principles of projective geometry applied to the straight line and conic . s. (c) corresponding chords ABand AB intersect on s. (d) tangents from S tSV=VN. Then the line parallel to s throughV is the vanishing line. The two columns above simply give a restatement of the sametheorem looked at from two slightly different points of view and forma good instance of the Principle of Duality. 80. Complete Inscribed Quadrangle and Complete Circum-scribed Quadrilateral of a Circle. Any four points A, B, C, B ona circle determine (1) three points S, T, U, asthe points of intersection of theirconnectors taken in pairs. Any four tangents a, b, c, d toa circle determine (1) three lines 6, t, u, as thelines joining the points of inter-sections of the tangents taken inpairs. 138 Principles of Projective Geonietrij (2) three linos 6-, t, a, as the (2) throe points >S, T, U, as connectors of the points S, T, U the intersections of s, t, u in pairs,in pairs. «. The lines s, t, ii, are the polarsof S, T, U with regard to the circle.(Art. 76.) Hence the tangentsa, b, c, d at the points A, B, G, Dintersect in pairs on s, t, u. As each side of the triangle8TU is the polar of the oppositevertex it is a self-polar or self-conjugate triangle. It follows that (a) The diagonal points triangle 8TU of the quadrangle ABGD isthe same as the diagonal triangle stu of the quadrilateral abed. The points 8, T, U, are thepoles of s, t, u. (Art. 76.) Hencethe points of contact A, B, G, Dare collinear in pairs with S, T, U. As each vertex of the trianglesta is the pole of the opposite sideit is a self-polar or self-conjugatetriangle. The Circle 139 (6) From the quadrilateral abed ac, bd, T, S] ad, be, U, aSJ- arc harmonic , ba, T, u\ (c) From the quadrangle A BOB , , S, r\ , , S, f/- are harmonic ranges. , , T, u\ (d) If the circle meets t in P, P and s in Q, Q P, P, S, U] , Q Q T u\ ^ ^■i^-nic ranges. In the figure, a does not
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Keywords: ., bookcentury1900, bookdecade1910, booksubjectgeometryprojective
