If BM and CN are the perpendiculars drawn on the sides AC and BC of the `Delta ABC`, prove that the points B, C, M and N are concyclic.

BE and CF are perpendicular on sides AC and AB of triangles ABC respectively . Prove that four points B , C ,E , F are concyclic. Also prove that the two angles of each of DeltaAEFandDeltaABC are equal.

In Delta ABC, AD and BE are perpendiculars from A and B to the sides BC and AC, then

In the adjoining figure, BM and CN are the altitudes drawn to the sides AC and AB respectively of DeltaABC . If BM=CN, prove that DeltaABC is isosceles.

In Figure,ABC is a triangle in which AB=AC. Point D and E are points onthe sides AB and AC respectively such that AD=AE .Show that the points B,C,E and D are concyclic.

ABC is an isosceles triangle in which AB=AC. If D and E are the mid-points of AB and AC respectively,prove that the points B,C,D and E are concyclic.

BM and CN are perpendiculars to a line passing through the vertex A of a triangle ABC. If Lis the mid-point of BC, prove that LM = LN.

AL, BM and CN are perpendicular from angular points of a triangle ABC on the opposite sides BC, CA and AB respectively. Delta is the area of triangle ABC, (r ) and R are the inradius and circumradius. Radius is the circum circle of Delta LMN is

In Delta ABC ,AD is a perpendicular drawn from A to the side BC. If AD^(2)=BD . CD , then prove that the triangle is a right-angled triangle.