MathDB

Problem 6, Olympic Revenge 2010

Source: IX Olympic Revenge - 2010

January 28, 2013

geometrygeometric transformationhomothetycircumcircleprojective geometrycomplex numbersgeometry proposed

Problem Statement

Let

ABC

to be a triangle and

\Gamma

its circumcircle. Also, let

D, F, G

and

E

, in this order, on the arc

BC

which does not contain

A

satisfying

\angle BAD = \angle CAE

and

\angle BAF = \angle CAG

. Let

D`, F`, G`

and

E`

to be the intersections of

AD, AF, AG

and

AE

with

BC

, respectively. Moreover,

X

is the intersection of

DF`

with

EG`

,

Y

is the intersection of

D`F

with

E`G

,

Z

is the intersection of

D`G

with

E`F

and

W

is the intersection of

EF`

with

DG`

.
Prove that

X, Y

and

A

are collinear, such as

W, Z

and

A

. Moreover, prove that

\angle BAX = \angle CAZ

.

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