MathDB

Let

ABC

be an equilateral triangle with the side length equals

a+ b+ c

. On the side

AB{}

of the triangle

ABC

points

C_1

and

C_2

are chosen, on the side

BC

points

A_1

and

A_2

, arc chosen, and on the side

CA

points

B_1

and

B_2

are chosen such that

A_1A_2 = CB_1 = BC_2 = a, B_1B_2 = AC_1 = CA_2 = b, C_1C_2 = BA_1 = AB_2 = c

. Let the point

A^{’}

be such that the triangle

A^{'} B_2C_1

is equilateral, and the points

A

and

A^{'}

lie on different sides of the line

B_2C_1

. Similarly, the points

B^{’}

and

C^{'}

are constructed (the triangle

B^{'} C_2A_1

is equilateral, and the points

B

and

B^{’}

lie on different sides of the line

C_2A_1

; the triangle

C^{'} A_2B_1

is equilateral, and the points

C

and

C^{'}

lie on different sides of the line

A_2B_1

). Prove that the triangle

A^{'}B^{'}C^{'}

is equilateral.

Problem Statement