Perpendiculars \(PX, PY, PZ\) are drawn from an arbitrary point \(P\) in the plane to the sides of the triangle \(ABC\); the circle \(XYZ\) cuts the sides again in \(X', Y', Z'\). Prove that the perpendiculars to the sides at \(X', Y', Z'\) are concurrent at, say, \(P'\). Show that, if \(Q, R\) are two points such that the feet of the four perpendiculars from \(Q, R\) to the sides \(AB, AC\) are concyclic, then \(\angle QAC = \angle RAB\). Hence, or otherwise, prove that \(P, P'\) have the property that their joins to any vertex of the triangle \(ABC\) are equally inclined to the sides through that vertex.
A line in space cuts a plane at \(P\) and is perpendicular to two distinct lines lying in the plane and passing through \(P\). Prove that it is perpendicular to every line in the plane. The perpendicular from one vertex of a tetrahedron to the opposite face cuts it in the orthocentre of that face. Prove the same to be true for the other vertices.
From a variable point on a diagonal \(WY\) of a parallelogram \(WXYZ\) lines are drawn through fixed points \(B', C'\) on \(XY, YZ\) to cut the opposite sides in \(Q, R\). Prove that \(QR\) is parallel to \(B'C'\). The mid-points of the sides of the triangle \(ABC\) are \(A', B', C'\) and the altitudes are the lines \(p, q, r\); the circumcentre, orthocentre, centroid and nine-point-centre are \(O, H, G\) and \(N\). By applying the above proposition to the parallelogram formed by \(q, r\) and the perpendicular bisectors of \(AC, AB\), or otherwise, prove that, if \(E\) is a variable point on \(OH\) and if \(A'E, B'E, C'E\) cut \(p, q, r\) respectively in \(P, Q, R\), then the triangle \(PQR\) is similar to the triangle \(ABC\). Investigate the ratio \(QR:BC\) in the cases where \(E\) is at \(O, H, G, N\).
Define the polar of a point \((x_1, y_1)\) with respect to the circle \[ a(x^2+y^2)+2gx+2fy+c=0, \] and find its equation. Interpret the equation of the polar geometrically in the cases (i) \(a=0\), and (ii) \(ac=f^2+g^2\); verify algebraically both these interpretations.
Prove that in general the locus of points whose tangents to a given conic \(S\) are perpendicular is a circle \(\Sigma\). If \(S\) is neither a parabola nor a rectangular hyperbola, prove that the tangents to \(S\) from a point \(A\) on \(\Sigma\) cut \(\Sigma\) again at opposite ends of a diameter of \(S\) which is conjugate to the diameter through \(A\).
Explain what is meant by the cross-ratio of four points (i) on a straight line, (ii) on a conic. \(A, B, C, D\) are four points of a conic \(\Gamma\). A line \(l\) through \(D\) meets \(BC, CA, AB\) respectively in \(P, Q, R\) and meets \(\Gamma\) again in \(S\). Prove that the cross-ratio \((ABCD)\) on \(\Gamma\) is equal to the cross-ratio \((PQRS)\) on \(l\).
\(H\) and \(K\) are two fixed points in the plane of a conic \(S\). Prove that the locus of a point \(P\) which moves so that \(PH\) and \(PK\) are conjugate with respect to \(S\) is a conic \(\Gamma\) through \(H\) and \(K\). Where does \(\Gamma\) meet \(S\)? The triangles \(ABC\) and \(A'B'C'\) are each self-polar with respect to \(S\). By considering pairs of conjugate lines through \(A\) and \(A'\), or otherwise, prove that \(A, B, C, A', B', C'\) lie on a conic.
Two conics \(S, S'\) are circumscribed to a triangle \(ABC\) and touch each other at \(A\). A line \(l\) through \(A\) meets \(S, S'\) again at \(P, P'\) respectively. Prove that the tangents to \(S\) and \(S'\) at \(P\) and \(P'\) meet in a point \(Q\) of \(BC\). Show further that the harmonic conjugate of \(l\) with respect to \(AB, AC\) meets \(S\) and \(S'\) again in two points, the tangents at which also meet in \(Q\).
A variable line \(lx+my+nz=0\) meets the conic \(S \equiv y^2-zx=0\) in two points \(P, P'\) such that the lines \(YP, YP'\) harmonically separate the fixed pair of lines \(ax^2+cz^2+2gzx=0\) through \(Y\). Show that the line \(PP'\) envelops a conic \(\Gamma\), and find the equation of \(\Gamma\) in tangential coordinates. Investigate conditions under which \(\Gamma\) may degenerate.
Three coplanar triangles \(A_1B_1C_1, A_2B_2C_2\) and \(A_3B_3C_3\) are such that \(B_1C_1, B_2C_2, B_3C_3\) meet at \(L\), \(C_1A_1, C_2A_2, C_3A_3\) meet at \(M\), and \(A_1B_1, A_2B_2, A_3B_3\) meet at \(N\), where \(L, M, N\) are collinear. Show that \(A_2A_3, B_2B_3, C_2C_3\) are concurrent (in a point \(O_1\), say). If \(O_2\) and \(O_3\) are similarly defined, show that \(O_1, O_2\) and \(O_3\) are collinear.