`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 a triangle ABC if angleA=theta and perpendiculars drawn from vertices B and C to respective opposite sides meet in P then find the value of angleBPC in terms of theta .

The lengths of the perpendiculars drawn from any point in the interior of an equilateral triangle to the respective sides are P_1 . P_2 and P_3 . The length of each side of the triangle is

The length of the perpendicular drawn from any point in the interior of an equilateral triangle to the respective sides are p_(1), p_(2) and p_(3) . The length of each side of the triangle is

In Delta P Q R , if P Q=Q R and L ,M and N are the mid-points of the sides P Q ,Q R and R P respectively. Prove that L N=M Ndot

O A and O B are fixed straight lines, P is any point and P M and P N are the perpendiculars from P on O Aa n dO B , respectively. Find the locus of P if the quadrilateral O M P N is of constant area.

Let O(0.0),A(6,0) and B(3,sqrt3) be the vertices of triangle OAB. Let R those points P inside triangle OAB which satisfy d(P, OA) minimum (d(P,OB),d(P,AB) where d(P, OA), d(P, OB) and d(P, AB) represent the distance of P from the sides OA, OB and AFB respectively. If the area of region R is 9 (a -sqrtb) where a and b are coprime, then find the value of (a+ b)

If the point A(0, 2) is equidistant from the points B(3, p) and C(p, 5), find p.