MathDB

Consider the equilateral triangle

ABC

with

|BC| = |CA| = |AB| = 1

. On the extension of side

BC

, we define points

A_1

(on the same side as B) and

A_2

(on the same side as C) such that

|A_1B| = |BC| = |CA_2| = 1

. Similarly, we define

B_1

and

B_2

on the extension of side

CA

such that

|B_1C| = |CA| =|AB_2| = 1

, and

C_1

and

C_2

on the extension of side

AB

such that

|C_1A| = |AB| = |BC_2| = 1

. Now the circumcentre of 4ABC is also the centre of the circle that passes through the points

A_1,B_2,C_1,A_2,B_1

and

C_2

. Calculate the radius of the circle through

A_1,B_2,C_1,A_2,B_1

and

C_2

.
[asy] unitsize(1.5 cm);
pair[] A, B, C;
A[0] = (0,0); B[0] = (1,0); C[0] = dir(60); A[1] = B[0] + dir(-60); A[2] = C[0] + dir(120); B[1] = C[0] + dir(60); B[2] = A[0] + dir(240); C[1] = A[0] + (-1,0); C[2] = B[0] + (1,0);
draw(A[1]--A[2]); draw(B[1]--B[2]); draw(C[1]--C[2]); draw(circumcircle(A[2],B[1],C[2]));
dot("

A

", A[0], SE); dot("

A_1

", A[1], SE); dot("

A_2

", A[2], NW); dot("

B

", B[0], SW); dot("

B_1

", B[1], NE); dot("

B_2

", B[2], SW); dot("

C

", C[0], N); dot("

C_1

", C[1], W); dot("

C_2

", C[2], E); [/asy]

Problem Statement