MathDB

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]

Problem Statement