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circles tangent to eachother and to the sides of triangle

Source: romania TST 5 / 2007 problem 2

June 13, 2007

geometrygeometric transformationprojective geometrygeometry proposedDesarguesMonge theorem

Problem Statement

Let

ABC

be a triangle, and

\omega_{a}

,

\omega_{b}

,

\omega_{c}

be circles inside

ABC

, that are tangent (externally) one to each other, such that

\omega_{a}

is tangent to

AB

and

AC

,

\omega_{b}

is tangent to

BA

and

BC

, and

\omega_{c}

is tangent to

CA

and

CB

. Let

\omega_{a}

be the common point of

\omega_{b}

and

\omega_{a}

,

\omega_{b}

the common point of

\omega_{c}

and

\omega_{a}

, and

F

the common point of

\omega_{a}

and

\omega_{b}

. Show that the lines

AD

,

BE

and

CF

have a common point.

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