We determined high-pressure melting curves for Fe3C, Fe7C3 and the Fe-Fe3C eutectic using laser-heated diamond anvil cell techniques. The principal criterion for melting is the observation of plateaus in the temperature vs. laser power function, which is an expected behavior at isobaric invariant points (e.g. congruent, eutectic, or peritectic melting) as increased power provides the latent heat of melting. We verified this technique by reproducing the melting curves of well-studied congruently melting compounds at high pressure (Fe, Pt, FeS, Pb), and by comparison with melting determinations made using thermocouple-based large-volume press techniques. The incongruent melting curve of Fe3C measured to 70 GPa has an apparent change in slope at ~ 8 GPa, which we attribute to stabilization of Fe7C3 at the solidus and the creation of a P-T invariant point. We observe that Fe7C3 melts at higher temperatures than Fe3C between 14 and 52 GPa and has a steep P-T slope, and on this basis predicts an expanding field of Fe7C3 + liquid with pressure. The Fe-Fe3C eutectic melting curve measured to 70 GPa agrees closely with multi-anvil data and thermodynamic calculations. We also measured the eutectic composition as a function of pressure using an in situ X-radiographic imaging technique, and find a rapid drop in carbon in the eutectic composition above about 20 GPa, generally consistent with previous thermodynamic calculations, and predict that the eutectic lies close to pure iron by ~ 50 GPa. We use these observations to extrapolate phase relations to core-relevant pressures. Convergence of the Fe3C and Fe-Fe3C eutectic melting curves indicate that Fe3C is replaced at the solidus by Fe7C3 at ~ 120 GPa, forming another P-T invariant point and a new eutectic between Fe and Fe7C3. Thus, Fe3C is unlikely to be an important crystallizing phase at core conditions, whereas Fe7C3 could become an important crystallizing phase.