ABC is an equilateral triangle with A(0,0) and B(a,0), `(a gt 0)`.
L,M and V are the foot of the perpendiculars drawn from a point P to the sides AB, BC, and CA, respectively. If P lies inside the triangle and satisfies the condition `PL^(2) = PM * PN,` then find the locus of P.

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

Find the coordinates of the foot of the perpendicular drawn from the point P(1,-2) on the line y = 2x +1. Also, find the image of P in the line.

ABC is a right angled triangle , right angled at C and P is the length of perpendicular from C on AB. If a, b and C are the length of sides BC, CA and AB respectively. Then___

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.

M is the foot of the perpendicular from a point P on a parabola y^2=4a x to its directrix and S P M is an equilateral triangle, where S is the focus. Then find S P .

Find the locus of the foot of the perpendicular from the point (a,0) to the straight line x-ty+at^(2)=0 , t being variable.

OX,OY and OZ are the perpendiculars drawn from an internal point O of the Delta ABC on its sides BC,CA and AB respectively. Prove that AZ^(2) + BX^(2) + CY ^(2) = AY^(2) + CX^(2) + BZ^(2)

In a right triangle ABC right-angled at B, if P and Q are points on the side AB and AC respectively, then___

Triangle ABC is isosceles with AB=AC and BC=65cm. P is a point on BC such that the perpendiculardistances from P to AB and AC are 24cm and 36cm, respectively. The area of triangle ABC (in sq cm is)

I is the incentre of the Delta ABC . The perpendicular drawn from I to BC intersects BC at. P. Prove that AB-AC=BP-CP .