# An Application of the Gergonne-Euler Theorem

An Application of the Gergonne-Euler Theorem

Let the intersection of lines and be denoted . Let the line through point P and parallel to the line be denoted by [P, ].

g

h

g⋂h

g

g

Let ABC and A'B'C' be two triangles. Let , , and . Let , , and . Then XX', YY', and ZZ' are concurrent.

X=BC⋂B'C'

Y=CA⋂C'A'

Z=AB⋂A'B'

X'=[A,BC]⋂[A',B'C']

Y'=[B,CA]⋂[B',C'A']

Z'=[C,AB]⋂[C',A'B']