\(ABC\) is a triangle; \(PQR\) is inscribed in \(ABC\), \(P\) lying on \(BC\), \(Q\) on \(CA\) and \(R\) on \(AB\). Prove that the circles \(AQR, BRP, CPQ\) have a common point, \(K\). Prove that all the triangles inscribed in \(ABC\), giving rise to this fixed point \(K\) are similar, and that the one of smallest area has vertices at the feet of the perpendiculars from \(K\) to the sides of \(ABC\). Outline a construction to find the smallest equilateral triangle inscribed in a given triangle.
From a thin uniform rod three lengths are cut and pinned together at their ends to form a triangular frame \(ABC\); \(J\) is the centre of gravity of the frame, \(I\) the incentre of \(ABC\) and \(G\) its centroid. Prove that \(J\) is the incentre of the triangle whose vertices are the mid-points of \(BC, CA, AB\); hence prove that \(G\) is a point of trisection of \(IJ\).
On the surface of a sphere, centre \(O\), are four points \(A, B, C, D\). Prove that \(AB\) is perpendicular to \(CD\) if and only if \[ \cos\angle AOC + \cos\angle BOD = \cos\angle AOD + \cos\angle BOC. \]
\(S\) is a circle and \(K\) a point outside it; \(\alpha\) is a given acute angle. Prove that there are precisely two pairs of points, \(Q, R\), such that \(Q\) and \(R\) are inverse points with respect to \(S\), \(KQ=KR\) and \(\angle QKR = 2\alpha\). \(\Sigma_1, \Sigma_2\) and \(\Sigma_3\) are three circles; \(\Sigma_2\) and \(\Sigma_3\) meet in two distinct points, both outside \(\Sigma_1\) and are not orthogonal. Prove that there are two positions for a point \(P_1\) such that its inverses, \(P_3, P_2\), with respect to \(\Sigma_2\) and \(\Sigma_3\) are themselves inverse points with respect to \(\Sigma_1\).
At points \(P, Q, R\) of a parabola tangents are drawn to make a triangle \(LMN\). Prove that the area of \(LMN\) is half the area of \(PQR\), and that the line joining the centroids of the two triangles is parallel to the axis of the parabola.
Prove that, if two rectangular hyperbolas intersect in four points \(A, B, C, D\), then any conic through \(A, B, C, D\) is a rectangular hyperbola. By reciprocating with respect to an arbitrary circle, or otherwise, prove that the director circles of a family of conics touching four fixed lines form a coaxial system.
Define an involution on a line. The six sides of a complete quadrangle cut a general line of the plane in six points; prove that the pairs of points arising from intersections with opposite sides of the quadrangle are pairs in an involution on the line. For what positions of the line does this cease to be true? Given a line and two pairs of points on it, \(P, P'\) and \(Q, Q'\), and a fifth point on it, \(R\), give a construction using only a ruler to find the point \(R'\) on the line which is paired with \(R\) under the involution which pairs \(P\) with \(P'\) and \(Q\) with \(Q'\).
(i) Two conics, \(S_1\) and \(S_2\), have double contact. \(P_1\) is a point which varies on \(S_1\) and the tangent to \(S_1\) at \(P_1\) cuts \(S_2\) at \(Q_2\) and \(R_2\); find the locus of the harmonic conjugate of \(P_1\) with respect to \(Q_2\) and \(R_2\). (ii) Two conics touch at \(A\) and cut at \(B\) and \(C\); their two common tangents, other than that at \(A\), intersect at \(X\). Prove that the tangent at \(A\) and \(AX\) cut \(BC\) at points which are harmonic conjugates with respect to \(B\) and \(C\).
A variable conic, \(S\), passes through the fixed points \(A, B, C\) and touches the fixed line \(l\), which does not contain \(A, B\) or \(C\). Prove that the locus of the pole of \(BC\) with respect to \(S\) is a conic inscribed in the triangle \(ABC\). Determine the points in which this locus meets \(l\).
On a conic, \(S\), are two points, \(A\) and \(B\); \(L\) is a variable point in the plane. \(AL, BL\) cut \(S\) again in \(Q, R\); \(AR, BQ\) meet in \(M\). Prove that, if \(L\) traces out a conic through \(A\) and \(B\), \(M\) traces out a conic through \(A, B\) and the other two points of intersection of \(S\) with the locus of \(L\). If the tangents at \(L\) and \(M\) to their respective loci meet at \(T\), prove the locus of \(T\) to be a straight line.