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A B C is an equilateral triangle with A(...

`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`

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To find the locus of the point \( P \) inside the equilateral triangle \( ABC \) given the condition \( PL^2 = PM \cdot PN \), we will follow these steps: ### Step 1: Define the Points and Triangle Given the vertices of the equilateral triangle: - \( A(0, 0) \) - \( B(a, 0) \) - \( C\left(\frac{a}{2}, \frac{a\sqrt{3}}{2}\right) \) ...
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