There was a little problem in my final exam that went "Show that a conformal equi-areal map is an isometry". I invoqued the caracterisation of "conformal" that the two metrics are proportional, say by a proportionality function L: E1=LE2, F1=LF2, G1=LG2. Then I invoked the caracterisation of "equiareal" that the determinants of the two metrics must be equal: E1G1-F1²=E2G2-F2². Combining these two proporties yields the equation E1G1-F1²=L²(E1G1-F1²) which means that L²=1, which means that L=+1 or -1. But for the map to be an isometry, it must be L=1 (the two metrics must be equal), so how do we reject L=-1?