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tangent nagelians with common point

Source: Sharygin 2023 - P17 (Grade-9-11)

March 4, 2023

geometrynagelianconcurrencytangent circlesSharygin Geometry OlympiadSharygin 2023

Problem Statement

A common external tangent to circles

\omega_1

and

\omega_2

touches them at points

T_1, T_2

respectively. Let

A

be an arbitrary point on the extension of

T_1T_2

beyond

T_1

, and

B

be a point on the extension of

T_1T_2

beyond

T_2

such that

AT_1 = BT_2

. The tangents from

A

to

\omega_1

and from

B

to

\omega_2

distinct from

T_1T_2

meet at point

C

. Prove that all nagelians of triangles

ABC

from

C

have a common point.

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