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The problem of determining a crack by overspecified boundary data is considered. When complete data are avaible on the external boundary.A link that is established between the Reciprocity gap functional and the Fourier transform of the temperature is introduced.If the crack is known (or assumed)to be line, an explicit inversion formulae is obtained and determination of the host line equation and the length of the crack in the two-dimensional (2D) situation. Numerical tests of the identification methods proposed show very good accuracies and significant computational costs. Keywords: Cauchy problem, Inverse geometric problem,the reciprocity gap principle,identification of crack, Fourier Transform,finite element method.

Line segment cracks identification via the reciprocity gap principle and Fourier transform