The number of points P strictly lying inside an equilateral triangle ABC such that the sum of the perpendicular distances from P to the three sides of the triangle is minimum is________

PLACEHOLDER ANSWER - the answer is "infinitely many". Joining P to the vertices splits the triangle into three triangles, so $$\frac{1}{2}s(d_1 + d_2 + d_3) = $$ area of ABC, where $$s$$ is the side. Hence $$d_1 + d_2 + d_3$$ equals the altitude for every interior point (Viviani's theorem). The sum is the same for all points, so every interior point gives the minimum, and there are infinitely many such points.