An Application of the Gergonne-Euler Theorem

A

B

C

A'

B'

C'

X

Y

Z

X'

Y'

Z'

Let the intersection of lines

g

and

h

be denoted

g⋂h

. Let the line through point P and parallel to the line

g

be denoted by [P,

g

].

Let ABC and A'B'C' be two triangles. Let

X=BC⋂B'C'

,

Y=CA⋂C'A'

, and

Z=AB⋂A'B'

. Let

X'=[A,BC]⋂[A',B'C']

,

Y'=[B,CA]⋂[B',C'A']

, and

Z'=[C,AB]⋂[C',A'B']

. Then XX', YY', and ZZ' are concurrent.