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.
Prove that the feet of the perpendiculars from the foci S, S' of an ellipse on a tangent lie on the auxiliary circle. Parallel lines SP, S'P' drawn towards the same parts meet the ellipse in P and P'. Prove that the tangents to the ellipse at P and P' meet on the auxiliary circle.
Prove that the equation \(x^2+y^2-2cx\sec\theta+c^2=0\) as \(\theta\) varies represents a system of coaxal circles with real limiting points L, L'. If P is any point on the circle \(\theta\) prove that PL:PL' = \(\tan\frac{\theta}{2}:1\), if L lies inside the circle.
Find the equation of the tangent at the point \((at^2, 2at)\) on the parabola \(y^2=4ax\). Find the equation of the circle circumscribing the triangle formed by the tangents to the parabola at the points whose parameters are \(t, t', t''\).