Equilateral triangles \(BCP, CAQ, ABR\) are drawn outward on the sides of an acute-angled triangle \(ABC\). Prove that the triangles \(ABC, PQR\) have the same centroid.
\(A, B, C, D\) are four distinct points on a given circle. A variable circle is drawn through \(B, C\) and another variable circle is drawn through \(A, D\). Prove that the radical axis of these variable circles cuts the given circle in pairs of points belonging to an involution. \newline Prove or disprove the following statement: ``The line joining two given corresponding points of the involution just obtained is the radical axis of two circles (one through \(B, C\) and the other through \(A, D\)) which are uniquely determined by those two given points.''
\(A, B, C, P\) are four non-coplanar points, and \(Q\) is a point of the line \(AP\). Prove that \(BP, CQ\) are skew and that \(BQ, CP\) are skew. \newline Through a point \(R\) of \(BC\) a line is drawn meeting \(PB\) in \(L\) and \(PC\) in \(M\); another line through \(R\) meets \(QB\) in \(L'\) and \(QC\) in \(M'\). Prove that \(LL', MM'\) meet on \(AP\).
The normal at a point \(P\) of an ellipse, of which \(S\) is a focus, meets the ellipse again in \(Q\), and the normal at \(Q\) meets the major axis in \(L\). A line through \(L\) parallel to \(PQ\) meets \(SP\) in \(R\). Prove that \(SQ=SR\).
The equation \[ ax^2+2hxy+by^2+2gx+2fy+c=0, \] referred to rectangular cartesian axes with their origin at \(O\), represents two straight lines meeting in \(A\). The circle on \(OA\) as diameter meets the diagonal other than \(OA\) of the parallelogram, which has the two given lines as sides and \(O\) as a vertex, in the points \(P, Q\). Obtain the equation of the line-pair \(OP, OQ\) in the form \[ (Gg-Ff)(x^2-y^2)+2(Gf+Fg)xy=0, \] where \(F=gh-af, G=hf-bg\).
The equations (referred to rectangular cartesian axes) \begin{align*} ax^2+2hxy+by^2-1&=0, \quad (a\neq b) \\ x^2+y^2-r^2&=0 \end{align*} represent a given conic and a given circle. Write down the equation of an arbitrary conic of the pencil determined by the given conic and the pair of tangents from an arbitrary point \(P(\xi, \eta)\) to the circle; and show that, when \(h=0\), this pencil includes a circle if, and only if, \(P\) is on one of the coordinate axes. Prove also the converse result that, if \(P\) is on one of the coordinate axes (not at the origin) and the pencil includes a circle, then \(h=0\). \newline Deduce that, in the general case, the equation of the principal axes of the conic is \[ \frac{x^2-y^2}{a-b} = \frac{xy}{h}. \]
Prove that through a given point \(P\) of a given parabola a unique circle can be drawn to osculate the parabola elsewhere. \newline Denoting the point of osculation by \(Q\), let the circle be drawn which passes through \(Q\) and touches the parabola at \(P\), the other point of intersection being \(R\). Prove that a circle can be drawn through \(P, R\) to touch the parabola at the mirror image of \(Q\) in the axis of the parabola. \newline [If you quote a substantial theorem about the intersections of a circle and a parabola you must prove it.]
The two lines \[ ax^2+2hxy+by^2=0 \quad (a>0, b>0) \] meet the rectangular hyperbola \(xy=c^2\) in the points \(P, Q\) lying in the first quadrant. Prove that, if the length of \(PQ\) is \(l\), then \[ \frac{l^2}{c^2} = -2 \left(\frac{1}{a}+\frac{1}{b}\right) \{h+\sqrt{(ab)}\}. \] Explain how the data ensure that the expression on the right-hand side is positive.
The general homogeneous coordinates of a point \(Q\) are \((\alpha, \beta, \gamma)\) with respect to a triangle of reference \(XYZ\) and unit point \(P(1, 1, 1)\). The lines \(XP, YP, ZP\) meet \(YZ, ZX, XY\) in \(L_1, M_1, N_1\) and the lines \(XQ, YQ, ZQ\) meet \(YZ, ZX, XY\) in \(L_2, M_2, N_2\). The line joining \(X\) to the point of intersection of \(M_1N_2, M_2N_1\) meets \(YZ\) in \(U\), and points \(V, W\) are similarly defined on \(ZX, XY\). Prove that the points \(U, V, W\) are collinear, lying on a line whose equation should be determined.
A conic passes through the vertex \(X\) of a triangle \(XYZ\) and meets \(XY, XZ\) in \(R, Q\) respectively. The line \(QR\) meets \(YZ\) in \(L\) and the tangent at \(X\) meets \(YZ\) in \(M\). Prove that \(L, M\) are pairs in the involution of which \(Y, Z\) form one pair and the points in which the conic meets \(YZ\) another. \newline Give a metrical interpretation of this result by taking the double points of that involution as the ``circular points at infinity.''