The vertex \(A\) of a triangle is at a fixed point of a given circle with centre \(O\). The base \(BC\) is a variable chord of the circle and passes always through a fixed point \(P\). Show that the locus of the orthocentre of \(ABC\) is a circle, and indicate its centre and radius in a sketch.
Assign a geometrical meaning to the expression \(x^2+y^2+2gx+2fy+c\). Establish the existence of a radical axis for a pair of circles. Show that the radical axes of three circles taken in pairs are concurrent.
A variable point \(P\) is taken on the ellipse \[ \frac{x^2}{a^2}+\frac{y^2}{b^2}=1, \] whose centre is \(O\), and the normal at \(P\) meets the diameter \(Q'OQ\) conjugate to \(OP\) in \(N\). Prove that the product \(PN \cdot OQ\) remains constant, and obtain the equation of the locus of \(N\).
Show that, referred to a pair of tangents as coordinates axes, the equation of a parabola may be written in the form \[ \sqrt{ax}+\sqrt{by}=1, \] and obtain the condition that the line \(lx+my=1\) shall touch the parabola. If \(\epsilon\) is the angle between the tangents forming the coordinate axes, show that the equation to the directrix is \[ (a\cos\epsilon+b)x+(a+b\cos\epsilon)y=\cos\epsilon. \] Find also the equation of the axis of the parabola.
A tetrahedron is such that two pairs of opposite edges are perpendicular. Show that the remaining pair must also be perpendicular. Prove that for such a tetrahedron each altitude passes through the orthocentre of the face to which it is drawn, and that the four altitudes are concurrent.
Show that referred to polar coordinates the equation of a conic may be written in the form \[ r(1+e\cos\theta)=l. \] The foci of a conic are \(S\) and \(H\), and from a point \(P\) of the curve focal chords \(PSQ\) and \(PHR\) are drawn meeting the curve again in \(Q\) and \(R\). Show that \(\dfrac{PS}{SQ}+\dfrac{PH}{HR}\) remains constant as \(P\) moves on the curve.
Establish Brianchon's theorem, that if a hexagon circumscribes a conic the joins of opposite vertices are concurrent. Hence prove that, if the tangent to a hyperbola at a point \(P\) meets the asymptotes in \(A\) and \(B\), then \(AP=PB\).
Prove that, if a variable conic touches four fixed straight lines, the locus of its centre is a straight line. A family of conics is such that its members have one focus and two tangents fixed. Prove that the auxiliary circles and the director circles of the members form two systems of coaxial circles, and that their radical axes are parallel.
A square \(PQRS\) of side \(x\) is inscribed in a triangle \(ABC\) in such a way that \(PQ\) lies on the side \(BC\) while \(R\) lies on \(CA\) and \(S\) on \(AB\). The sides of similar squares associated with \(CA\) and \(AB\) are of lengths \(y\) and \(z\). Prove that \[ x^{-1}+y^{-1}+z^{-1} = a^{-1}+b^{-1}+c^{-1}+r^{-1}, \] where \(a, b, c\) are the lengths of the sides of the triangle and \(r\) is the radius of its inscribed circle.
Prove that \[ 2^{-n}\sin\theta\operatorname{cosec}(\theta/2^n) = \cos(\theta/2)\cos(\theta/2^2)\dots\cos(\theta/2^n), \] \(n\) being an integer. Obtain the limit of the right-hand side when \(n\) is increased indefinitely, and hence establish the infinite product relation for \(\pi\) \[ \frac{2}{\pi} = \frac{\sqrt{2}}{2} \cdot \frac{\sqrt{2+\sqrt{2}}}{2} \cdot \frac{\sqrt{2+\sqrt{2+\sqrt{2}}}}{2} \dots. \]