MathDB

There are three circles

w_1, w_2, w_3

. Let

w_{i+1} \cap w_{i+2} = A_i, B_i

, where

A_i

lies insides of

w_i

. Let

\gamma

be the circle that is inside

w_1,w_2,w_3

and tangent to the three said circles at

T_1, T_2, T_3

. Let

T_iA_{i+1}T_{i+2}

's circumcircle and

T_iA_{i+2}T_{i+1}

's circumcircle meet at

S_i

. Prove that the circumcircles of

A_iB_iS_i

meet at two points. (

1 \le i \le 3

, indices taken modulo

3

)
If one of

A_i,B_i,S_i

are collinear - the intersections of the other two circles lie on this line. Prove this as well.

Problem Statement