If \(P\) is a point on the circumcircle of a triangle \(ABC\) and \(L\), \(M\), \(N\) are the feet of perpendiculars from \(P\) to the sides, show that \(LMN\) is a straight line (the pedal line of \(P\)). Prove that if \(H\) is the orthocentre, the pedal line of \(P\) bisects \(PH\).
Establish the existence of the radical axis of a pair of circles. Show how to construct the radical axis of two non-intersecting circles. Prove that for the three pairs of circles taken from three given circles the radical axes are concurrent.
Find the equation of the normal at the point \((a\cos\phi, b\sin\phi)\) of the ellipse $$b^2x^2 + a^2y^2 = a^2b^2.$$ Show that if \(\alpha = x\), \(\beta\), \(\gamma\) are three different points on the ellipse, the normals at them will be concurrent if $$\sin(\beta+\gamma)+\sin(\gamma+\alpha)+\sin(\alpha+\beta) = 0.$$ Hence show that if the normals at four different points \(\alpha\), \(\beta\), \(\gamma\), \(\delta\) are concurrent then \(\alpha + \beta + \gamma + \delta\) is an odd multiple of \(\pi\).
The equation of a conic (referred to Cartesian or homogeneous coordinates) is denoted by \(S = 0\), and the tangent at a general point \(P\) of it has equation \(T = 0\), while a line through \(P\) is denoted by \(L = 0\). Interpret the equations:
Show that the polar equation to a conic, referred to a focus as pole, has the form $$\frac{l}{r} = 1 + e\cos(\theta - \gamma),$$ where \(\gamma\) is the angular position of the nearer apse. Find the polar equation of the tangent at the point \(\theta = \alpha\). A second conic, for which \(\gamma = 0\), has the same focus and eccentricity but different major axis and is such that the two curves touch at \(\theta = \alpha\). Determine the semi-latus rectum of the second conic. What happens if \(e \geq 1\)?
Find the tangential equation of the conic whose equation referred to rectangular axes is $$ax^2 + 2hxy + by^2 + 2gx + 2fy + c = 0.$$ Show that the focuses of the conic are at the intersections of the two rectangular hyperbolas $$Cx^2 - Cy^2 - 2Gx + 2Fy + A - B = 0,$$ $$Cxy - Fx - Gy + H = 0,$$ where \(A\), \(B\), \(C\), \(F\), \(G\), \(H\) are the coefficients in the tangential equation. Hence, or otherwise, find the real focuses of the conic $$x^2 + 12xy - 4y^2 - 6x + 4y + 9 = 0.$$ Give also the equations of the corresponding directrices.
The pair of tangents from the point \((2, 1, 1)\) of the conic \(y^2 = 2x\) to the conic \(ax^2 + by^2 + cz^2 = 0\) meet the first conic again in points \(P\) and \(Q\). Determine the equation of the line \(PQ\). Show that if \(ac + b^2 = 0\) there are an infinite number of triangles inscribed to the first conic and circumscribed to the second.
Show that the double points of the involution determined by the two pairs of points $$ax^2 + 2bx + c = 0 \quad \text{and} \quad a'x^2 + 2b'x + c' = 0,$$ where \(x\) is a homographic coordinate on a line, are given by $$(ab' - a'b)x^2 - (ca' - c'a)x + (bc' - b'c) = 0.$$ Three points \(x_i\) (\(i = 1, 2, 3\)) on a line satisfy $$ax^3 + 3bx^2 + 3cx + d = 0, \quad (ad \neq bc),$$ and the harmonic conjugate of \(x_i\) with respect to the other pair of points is denoted by \(x_i'\). Show that the pair of points \(x_i\) and \(x_i'\) satisfy the quadratic equation $$(ax_i + b)x^2 + 2(bx_i + c)x + (cx_i + d) = 0.$$ Deduce that the three pairs of points \(x_i\), \(x_i'\) for \(i = 1, 2\), and \(3\) are in involution.
Prove that \(\sum_{r=1}^{4} \cos^4 r\pi/9 = 19/16\). Find also the numerical value of \(\sum_{1}^{4} \sec^4 r\pi/9\).
A closed polygon of \(2n\) sides, \(n\) of which are of length \(a\) and \(n\) of length \(b\), is inscribed in a circle. Show that the radius of the circle is independent of the arrangement of the sides, and find its value.