MathDB

Triangles

ABC

and

DEF

, having a common incircle of radius

R

, intersect at points

X_1, X_2, \ldots , X_6

and form six triangles (see the figure below). Let

r_1, r_2,\ldots, r_6

be the radii of the inscribed circles of these triangles, and let

R_1, R_2, \ldots , R_6

be the radii of the inscribed circles of the triangles

AX_1F, FX_2B, BX_3D, DX_4C, CX_5E

and

EX_6A

respectively. https://i.ibb.co/BspgdHB/Image.jpg Prove that

\sum_{i=1}^{6} \frac{1}{r_i} < \frac{6}{R}+\sum_{i=1}^{6} \frac{1}{R_i}

U. Maksimenkau

Problem Statement