. The principles of projective geometry applied to the straight line and conic . CM^ — :i constant Hence the locus of N is a circle whose centre is at M. 15. If two vertices of a quadrilateral subtend at a gioen point an angle equalto the angle which the two opposite vertices subtend at the same point, the feet of theperpendiculars from the given point on the sides of the quadrilateral are concyclic:. Let three of the sides of the quadrilateral be lines BL\ CA, AB and let thefourth side meet these lines in A, B, C respectively. Let the feet of the perpen-diculars from any point P on BC, CA, AB
. The principles of projective geometry applied to the straight line and conic . CM^ — :i constant Hence the locus of N is a circle whose centre is at M. 15. If two vertices of a quadrilateral subtend at a gioen point an angle equalto the angle which the two opposite vertices subtend at the same point, the feet of theperpendiculars from the given point on the sides of the quadrilateral are concyclic:. Let three of the sides of the quadrilateral be lines BL\ CA, AB and let thefourth side meet these lines in A, B, C respectively. Let the feet of the perpen-diculars from any point P on BC, CA, AB and BC be K, L, i/and N respectively. 358 Principles of Projective Geometry Then the angle NLK==NLP- FLK= NBP- PCK and the angle NMK= NMP - PMK = NCP - PBK. Therefore NLK- NMK= {NBP- NCP) - {PCK- PBK) = BPC-CPB. Hence, if the angles BPC and CPB are equal, the angles NLK and NMK arealso equal and the points A, L, M, N are concyclic. EXAMPLES. (1) If AM, YY, ZZ be the diagonals of a quadrilateral circumscribing a circle,Z, J/, N the middle points, and 0 the centre of the circle, then OX. OX oz. oz OL OM ON (2) If from the vertices of a triangle perpendiculars be drawn to any straightline, and from the feet of these perpendiculars three perpendiculars are drawn onthe opposite sides of the triangle, prove that these perpendiculars are concurrent. Use Addendum 5. (3) If lines be drawn from
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