\(ABC\) is a triangle, \(S\) its inscribed circle, and \(S_1\), \(S_2\), \(S_3\) the three escribed circles. Show that the remaining common tangent of \(S\), \(S_1\) and the remaining common tangent of \(S_2\), \(S_3\) are parallel to a side of the pedal triangle of \(ABC\) (i.e. the triangle formed by the feet of the altitudes).
Circles are drawn through a fixed point \(A\) to cut a fixed line \(l\), not passing through \(A\), at a fixed angle \(\theta\). Show (by inversion or otherwise) that, in general, these circles envelop a circle for which \(A\) and its image in \(l\) are inverse points. Examine the case \(\theta = \frac{1}{4}\pi\).
\(ABC\) is the triangle formed by the tangents to the circle \(x^2 + y^2 = r^2\) at the points \((r\cos\theta, r\sin\theta)\), for \(\theta = \alpha\), \(\beta\), \(\gamma\). (It is to be assumed that the triangle is a proper one.) Prove that the coordinates of the centroid of the triangle are $$\left( \frac{r}{12D} \sum \cos\alpha[3 + \cos(\beta - \gamma)], \quad \frac{r}{12D} \sum \sin\alpha[3 + \cos(\beta - \gamma)] \right),$$ where \(D = \cos\frac{1}{2}(\beta - \gamma) \cos\frac{1}{2}(\gamma - \alpha) \cos\frac{1}{2}(\alpha - \beta)\). Verify that the point $$\left( \frac{r}{4D} \sum \cos\alpha[1 + \cos(\beta - \gamma)], \quad \frac{r}{4D} \sum \sin\alpha[1 + \cos(\beta - \gamma)] \right)$$ is the orthocentre of the triangle. Prove that the centre of a circle touching the three sides of a triangle lies on the line joining the orthocentre and centroid if, and only if, the triangle is isosceles.
A variable tangent to a parabola meets the tangents at two fixed points \(P\), \(Q\) in \(A\) and \(B\). Prove that, in general, the circle on \(AB\) as diameter envelops a conic, which touches the tangents at \(P\) and \(Q\) to the parabola at the points where they meet the directrix.
Two circles are drawn, each touching an ellipse in two points, and touching each other. If the eccentricity of the ellipse is \(1/\sqrt{2}\), prove that the sum of the squares of the radii of the circles is equal to the square of the minor semi-axis of the ellipse.
Show that the locus of centres of circles touching two given circles is a pair of conics. Discuss which type these are (ellipses, hyperbolas, or other), discriminating between possible relative positions of the circles, and equality or inequality of their radii.
Sketch the curve \(x^4 + y^4 - 2x^2 a = 0\) for the values 2, 1, \(\frac{1}{4}\), 0, \(-1\) of the parameter \(a\). A tetrahedron has the property that any two opposite edges are perpendicular. Prove that the line joining this point to the mid-point of any edge of the tetrahedron is equal and parallel to the line joining the mid-points of the opposite edge to the circumcentre of the tetrahedron.
\(X\), \(S\) are opposite ends of the diameter of a circle \(C\) and \(l\) is the line tangent to \(C\) at \(N\), \(P\) is a variable point on \(C\) other than \(N\), and \(NP\) cuts \(l\) in \(P'\). Prove that there is a fixed circle with respect to which \(P\) and \(P'\) are inverse points. State, without proof, what are the figures into which a plane and a sphere invert with respect to a sphere. \(X\), \(S\) are opposite ends of a diameter of a sphere \(Q\) and \(p\) is the plane tangent to \(Q\) at \(S\). The projection \(P'\), of a point \(P\) of \(Q\) is defined as the point in which \(NP\) meets \(p\). Prove that the projection of a circle of \(Q\) is a circle or straight line of \(p\), and characterise the circles of \(Q\) whose projections are straight lines.
A man is unwinding a string wrapped round a smooth closed convex curve \(ABCD\) on a piece of paper. When \(AB\) (of length \(a\)) is unwound, that part of the string becomes impregnated with ink. Prove that when \(ABC\) has been unwound, the area of the ink blot is \(\pi \int_B^C s \, d\psi\), where \(s\) is the arc-length measured from the mid-point of \(AB\), and \(\psi\) is the angle between the tangent and some fixed direction.
Define exactly what is meant by the derivative \(dy/dx\) of a function \(y = f(x)\). Obtain from first principles the derivatives of