The failure envelope of such a test is indeed a straight line of the form : t=c+sigma*tan(phi) where t : shear strength c : cohession sigma : normal stress on failure surface phi : friction angle
All the above parameters are expressed in terms of total stresses. In order to visualize why this is a straight line have in mind that the test procedure is done 3 times. Each time in a specimen of soil you apply a confining pressure allowing it to consolidate. At the end of consolidation an axial load is applied without allowing any change in water content. At failure the minor principal stress sigma3 = the confining pressure and the major principal stress sigma1 = the vertical stress are known thus you can draw the Mohr circle of failure. For the 3 tests sigma1,1<sigma1,2<sigma1,3 and sigma3,1 < sigma3,2 < sigma3,3.
The failure envelope does not pass through the peak point on a Mohr circle but is rather tangent to the three circles.