MathDB

Let points

A_1

A_2

and

A_3

lie on the circle

\Gamma

in a counter-clockwise order, and let

P

be a point in the same plane. For

i \in \{1,2,3\}

, let

\tau_i

denote the counter-clockwise rotation of the plane centred at

A_i

, where the angle of rotation is equial to the angle at vertex

A_i

\triangle A_1A_2A_3

. Further, define

P_i

to be the point

\tau_{i+2}(\tau_{i}(\tau_{i+1}(P)))

, where the indices are taken modulo

3

(i.e.,

\tau_4 = \tau_1

and

\tau_5 = \tau_2

).
Prove that the radius of the circumcircle of

\triangle P_1P_2P_3

is at most the radius of

\Gamma

.
Proposed by Anant Mudgal

5