A triangle ABC and a point P inside this triangle are given.
Define D,E and F as the midpoints of AP,BP and CP, respectively. Furthermore, let R be the intersection of AE and BD,S the intersection of BF and CE, and T the intersection of CD and AF.
Prove that the area of hexagon DRESFT is independent of the position of P inside the triangle.[asy]
unitsize(1 cm);pair A, B, C, D, E, F, P, R, S, T;A = (0,0);
B = (5,0);
C = (1.5,4);
P = (2,2);
D = (A + P)/2;
E = (B + P)/2;
F = (C + P)/2;
R = extension(A,E,B,D);
S = extension(B,F,C,E);
T = extension(C,D,A,F);draw(A--B--C--cycle);
draw(A--P);
draw(B--P);
draw(C--P);
draw(A--F--B);
draw(B--D--C);
draw(C--E--A);dot("A", A, SW);
dot("B", B, SE);
dot("C", C, N);
dot("D", D, dir(270));
dot("E", E, dir(270));
dot("F", F, W);
dot("P", P, dir(270));
dot("R", R, dir(270));
dot("S", S, SW);
dot("T", T, SE);
[/asy] geometryareahexagonmidpointsareas