Question

What is arbitary point of an equilateral triangle?

Answer 1

Answer:

is a well-known fact (with proof sketched in the comments below the question) that

|PD|+|PE|+|PF|=height of triangle

so that the numerator of the target ratio is independent of P. It is perhaps less-well-known that

|AF|+|BD|+|CE|=|FB|+|DC|+|EA|=semi-perimeter of triangle(⋆)

Step-by-step explanation:

Ortha's Theorem. Orthians at points D, E, F on respective sides BC, CA, AB of △ABC concur if and only if

|AF|2+|BD|2+|CE|2=|FB|2+|DC|2+|EA|2