In a triangle
![ABC,](/media/m/8/a/f/8afcbd6e815ca10256c79a5b310e3d67.png)
choose any points
![K \in BC, L \in AC, M \in AB, N \in LM, R \in MK](/media/m/b/a/2/ba229ce88e30f0c72e79eae967ee2962.png)
and
![F \in KL.](/media/m/9/5/8/958785ef4bdbc92120f825ce5f9f89cd.png)
If
![E_1, E_2, E_3, E_4, E_5, E_6](/media/m/5/1/2/5128a420d1769587938ab1a2fa17486e.png)
and
![E](/media/m/8/b/0/8b01e755d2253cb9a52f9e451d89ec11.png)
denote the areas of the triangles
![AMR, CKR, BKF, ALF, BNM, CLN](/media/m/1/b/9/1b9fa97045269a1e3e0351b360f6f107.png)
and
![ABC](/media/m/a/c/7/ac75dca5ddb22ad70f492e2e0a153f95.png)
respectively, show that
%V0
In a triangle $ABC,$ choose any points $K \in BC, L \in AC, M \in AB, N \in LM, R \in MK$ and $F \in KL.$ If $E_1, E_2, E_3, E_4, E_5, E_6$ and $E$ denote the areas of the triangles $AMR, CKR, BKF, ALF, BNM, CLN$ and $ABC$ respectively, show that
$$E \geq 8 \cdot \sqrt [6]{E_1 E_2 E_3 E_4 E_5 E_6}.$$