The angular points of a rectangle A, B, C, D are the middle points of the sides of a plane quadrilateral of which the lengths of two opposite sides are given. Construct the quadrilateral.
Construct the common tangents to two given circles. The radical axis of two circles external to each other intersects the external common tangents in H, H' and an internal common tangent in K, prove that KH.KH' is equal to the rectangle contained by the radii of the circles.
Prove that the polar reciprocal of one circle with respect to another is a conic, and find the position of its asymptotes and foci. P, P' are points on the auxiliary circle of an ellipse at the extremities of a diameter. The lines from P to the foci of the ellipse when produced meet the circle in T and T' respectively. Prove that P'T, P'T' are tangents to the ellipse.
Prove that if corresponding sides of two coplanar triangles meet in three collinear points, their corresponding vertices lie on three concurrent straight lines. If the corresponding edges of two tetrahedra ABCD and A'B'C'D' intersect in six coplanar points, prove that corresponding vertices lie on four concurrent straight lines.
Prove that the circles \(x^2+y^2-2\lambda x - c^2=0\), as \(\lambda\) varies, form a coaxal system. Find the equation of the system of circles that cut these circles orthogonally. Tangents parallel to the line \(x \sin\theta - y\cos\theta = 0\) are drawn to the system of circles \[ x^2+y^2-2\lambda x-c^2=0. \] Prove that the locus of their points of contact is \[ x^2+2xy\tan\theta-y^2+c^2=0. \]
ABCD is a parallelogram and E is any point in the diagonal BD. DF drawn parallel to AE meets AC in F. Prove that BF is parallel to EC.
Circles PAQ and PBQ intersect in P and Q and the tangents at A and B are parallel. PA intersects the circle PBQ in R, BR intersects the tangent at A in C. Prove that the quadrilateral ABCQ is cyclic.
Define the "nine-points" circle of a triangle and prove the property from which it derives its name. Given the inscribed and circumscribed circles of a triangle in position, prove that the orthocentre lies on a fixed circle.
Define the polar of a point with respect to a circle. Prove that if the polar of A passes through B then the polar of B passes through A. Given two pairs of conjugate points with respect to a circle, find the locus of the centre of the circle.
If the lines joining corresponding vertices of two triangles are concurrent prove that the points of intersection of corresponding sides are collinear. Prove also that the six points in which non-corresponding sides intersect lie on a conic.