In equilateral triangle ABC with interior point D, if the perpendicular distances from D to the sides of 4,5, and 6, respectively, are given, then find the area of `triangleA B Cdot`

On the outside of the equilateral triangle ABC and in the angular region angleABC , O is a point. OP, OQ and OR are the perpendiculars drawn from O to the sides. AB, BC and CA respectively. Prove that the height of the triangle = OP + OQ - OR.

If circumradius of triangle ABC is 4 cm, then prove that sum of perpendicular distances from circumcentre to the sides of triangle cannot exceed 6 cm

A B C is an equilateral triangle with A(0,0) and B(a ,0) , (a>0). L, M and N are the foot of the perpendiculars drawn from a point P to the side A B ,B C ,a n dC A , respectively. If P lies inside the triangle and satisfies the condition P L^2=P MdotP N , then find the locus of Pdot

In triangle A B C , the equation of the right bisectors of the sides A B and A C are x+y=0 and y-x=0 , respectively. If A-=(5,7) , then find the equation of side B Cdot

The centroid of a triangle ABC is at the point (1,1,1) if the coordinates of A and B are (3,-5,7) and (-1,7,-6) respectively find the coordinates of the point C

If the perpendicular distances of the sides of an equilateral triangle from any point within the triangle be 10 cm, 12 cm and 18 cm, then find the area of the triangle.

The equations of the perpendicular bisectors of the sides A Ba n dA C of triangle A B C are x-y+5=0 and x+2y=0 , respectively. If the point A is (1,-2) , then find the equation of the line B Cdot

If the coordinates of the vertices of triangle A B C are (-1,6),(-3,-9) and (5,-8) , respectively, then find the equation of the median through Cdot

Let ABC be an acute angled triangle with orthocenter H.D, E, and F are the feet of perpendicular from A,B, and C, respectively, on opposite sides. Also, let R be the circumradius of DeltaABC . Given AH.BH.CH = 3 and (AH)^(2) + (BH)^(2) + (CH)^(2) = 7 Then answer the following Value of R is

Perpendicular are drawn from an internal point of an equilateral triangle to its sides. If the lengths of the perpendiculars be 9 cm, 11 cm and 10 cm, then find the sides and area of the triangle