Problem 3

Let three circles

\Gamma_1,\Gamma_2,\Gamma_3

, which are non-overlapping and mutually external, be given in the plane. For each point

P

in the plane, outside the three circles, construct six points

A_1,B_1,A_2,B_2,A_3,B_3

as follows: for each

i=1,2,3

A_i,B_i

are distinct points on

\Gamma_i

such that the lines

PA_i

and

PB_i

are both tangents to

\Gamma_i

. Call

P

exceptional if the three lines

A_1B_1,A_2B_2,A_3B_3

are concurrent. Show that every exceptional point of the plane, if any exists, lies on the same circle.

Step 3 of 5: Apply similarity

\triangle PA_iO_i\sim\triangle A_iX_iO_i\Longrightarrow O_iX_i\cdot O_iP=r_i^2

Detailed analysis

Triangles PA_iO_i and A_iX_iO_i are right triangles and are similar. Therefore O_iX_i/O_iA_i=O_iA_i/O_iP, so O_iX_i·O_iP=O_iA_i^2=r_i^2.