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An Application of the Gergonne-Euler Theorem

Let the intersection of lines

and

be denoted

g⋂h

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

be denoted by [P,

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.

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