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

In triangle ABC, base BC is fixed. Then prove that the locus of vertex A such that tan B+tan C= Constant is parabola.

The base of a triangle is divided into three equal parts. If t_1, t_2,t_3 are the tangents of the angles subtended by these parts at the opposite vertex, prove that (1/(t_1)+1/(t_2))(1/(t_2)+1/(t_3))=4(1+1/t_2^2)dot

Prove that the locus of the center of the circle which touches the given circle externally and the given line is a parabola.

Tangents are drawn to the parabola at three distinct points. Prove that these tangent lines always make a triangle and that the locus of the orthocentre of the triangle is the directrix of the parabola.

If the base angles of triangle are ((22 1/2))^@ and (112 1/2)^@ , then prove that the altitude of the triangle is equal to 1/2 of its base.

Prove that the locus of the point that moves such that the sum of the squares of its distances from the three vertices of a triangle is constant is a circle.

If a triangle and a parallelogram lie on the same base and between the same parallels, then prove that the area of the triangle is equal to half of the area of parallelogram.

Let alt=blt=c be the lengths of the sides of a triangle. If a^2+b^2 < c^2, then prove that triangle is obtuse angled .