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Turkey NMO 2012 Problem 6

Source: Turkey NMO 2012

November 26, 2012

geometrycircumcircletrigonometrygeometry proposed

Problem Statement

Let

BF

and

D

be points on segments

[AE]

and

[AF]

respectively. Excircles of triangles

ABF

and

ADE

touching sides

BF

and

DE

is the same, and its center is

I

.

BF

and

DE

intersects at

C

. Let

P_1, P_2, P_3, P_4, Q_1, Q_2, Q_3, Q_4

be the circumcenters of triangles

IAB, IBC, ICD, IDA, IAE, IEC, ICF, IFA

respectively.
a) Show that points

P_1, P_2, P_3, P_4

concylic and points

Q_1, Q_2, Q_3, Q_4

concylic. b) Denote centers of theese circles as

O_1

and

O_2

. Prove that

O_1, O_2

and

I

are collinear.

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