MathDB

Let

\Omega

be the circumcircle of the triangle

ABC

. The points

M_a,M_b

and

M_c

are the midpoints of the sides

BC, CA

and

AB{}

, respectively. Let

A_l, B_l

and

C_l

be the intersection points of

\Omega

with the rays

M_cM_b, M_aM_c

and

M_bM_a

respectively. Similarly, let

A_r, B_r

and

C_r

be the intersection points of

\Omega

with the rays

M_bM_c, M_cM_a

and

M_aM_b

respectively. Prove that the mean of the areas of the triangles

A_lB_lC_l

and

A_rB_rC_r

is not less than the area of the triangle

ABC

.
Proposed by L. Shatunov and T. Kazantseva

Problem Statement