SELECTION TESTS FOR THE 2019 BMO AND IMO

Question

Let ABC be a non-isosceles triangle, and let I be its incenter. Let _A be the circle through I, tangent to AB and AC and crossing the segment AI, let _B be the circle through I tangent to BC and BA and crossing the segment BI, and let _C be the circle through I, tangent to CA and CB and crossing the segment CI. The circles _B and _C cross again at A', the circles _C and _A cross again at B', and the circles _A and _B cross again at C'. Prove that the circles AA'I, BB'I and CC'I cross again at a point different from I. IMO 1997, Longlist

Accepted Answer

Let the internal bisectrices of the angles

B A C

,

C B A

and

A C B

cross the circle

A B C

again at

M_{A}

,

M_{B}

and

M_{C}

, respectively, and let

γ_{A}

,

γ_{B}

and

γ_{C}

be centred at

O_{A}

,

O_{B}

and

O_{C}

, respectively; clearly,

O_{A}

,

O_{B}

and

O_{C}

lie on the segments

A I

,

B I

and

C I

, respectively.

The centre of the circle

A A^{'} I

is the point where the perpendicular bisectrix of the segment

A I

crosses the perpendicular bisectrix of the segment

A^{'} I

. It is a fact that the former is the line

M_{B} M_{C}

; the latter is, of course, the line

O_{B} O_{C}

, so the lines

M_{B} M_{C}

and

O_{B} O_{C}

cross at the centre of the circle

A A^{'} I

. Similarly, the lines

M_{C} M_{A}

and

O_{C} O_{A}

cross at the centre of the circle

B B^{'} I

, and the lines

M_{A} M_{B}

and

O_{A} O_{B}

cross at the centre of the circle

C C^{'} I

.

Finally, since the triangles

M_{A} M_{B} M_{C}

and

O_{A} O_{B} O_{C}

are in perspective from

I

, the conclusion follows by Desargues' theorem.