Having given the bases and the sum of the areas of a number of triangles which have a common vertex, show that the locus of the vertex is a straight line.

If on a given base B C , a triangle is described such that the sum of the tangents of the base angles is m , then prove that the locus of the opposite vertex A is a parabola.

If the base of a triangle and the ratio of tangent of half of base angles are given, then identify the locus of the opposite vertex.

If on a given base BC[B(0,0) and C(2,0)], a triangle is described such that the sum of the base angles is 4, then the equation of the locus of the opposite vertex A is parabola whose directrix is y=k. The value of k is _________ .

If the bases of two triangles are situated on a same line and the other vertex of the two triangles are common, then the ratio of the areas of two triangles are _________ to the ratio of their bases

If the bases of two triangles are situated on same line and the other vertex of the two triangles are common, then the ratio of the areas of two triangles are ______to the ratio of their bases.

The base of an isosceles triangle measures 4 units base angle is equal to 45^(@) . A straight line cuts the extension of the base at a point M at the angle theta and bisects the lateral side of the triangle which is nearest to M. The area of quadrilateral which the straight line cuts off from the given triangle is

Let A (2,3) and B(-2, 1) be vertices of a triangle ABC. If the centroid of this triangle moves on the line 2x+3y=1, then the locus of the vertex C is the line-

If the equation of the base of an equilateral triangle is 2x - y=1 and the vertex is (-1,2), then the length of a side of the triangle is-

If the equation of base of an equilateral triangle is 2x-y=1 and the vertex is (-1,2), then the length of the sides of the triangle is