The fall of Rome in the fifth century, the empire which had dominated Europe for the last half millennium dissolving and giving way to dozens of European kingdoms is what started the Middle ages. Not one man taking over some English or French kingdom..

The end is a bit vague but because of the Roman beginning, I like the fall of Constantinople/the Eastern Roman Empire as the end of the middle ages. The Turks breaking through Constantinople's walls with cannon marked an end to the supremacy of castles.

I think it ended in Agincourt at the battle of Agincourt where projectile weapons were more valuable than heavy infantry.

Except that isn't true. Actually I'd say the opposite is true, by the end of the Hundred Years' war armour had improved to the point that heavily armoured men were practically impenetrable to arrows and shields started to fall out of favour because they weren't necessary against arrows anymore. The French lost at Agincourt because they marched heavy laden men across a field of mud. Reaching the enemy lines too tired to fight properly and thinned by people simply falling, being unable to get up and drowning in the mud.

For my personal use, I define the Middle Ages ("Medieval" period and the "dark ages") as beginning when the Vandals sacked Rome in 455 and ends when Mehmet II captures Constantinople in 1453.

In addition to being just shy of exactly a thousand years, this dating also appeals to me because both events mark a very profound change. 455 is the death knell of the Western Roman Empire, which had stumbled along and could possibly have been saved right up until that point. It marks the ascent of the Eastern Roman Empire -- and 1453, the end date, marks said empire's death. The Byzantines had ups and downs across the thousand years in between (and, at times, they were de facto Ottoman vassals), true, but 1453 put an end to all that. It also marked the death of the last vestiges of the Roman Empire, marked the end of conventional siege warfare, and marked the beginning of what would soon become widespread firearm usage in Europe.

The other possible end date I've considered is one related to Martin Luther (probably his letter to the archbishop), but that's a rather late date and is further complicated by the fact that it places the end of the Middle Ages long after Cosimo de' Medici (who was contemporaneous with Mehmet II) and even his grandson, Lorenzo de' Medici, would be dead long before that. Luther's letter was sent, if memory serves, in 1517... but that would mean the Middle Ages includes most of the lives of da Vinci and Machiavelli, which is just a little strange. By moving the date forward to 1453, you let these rather famous champions of the Italian Renaissance kick off their own particular period in history without stepping on the medieval period's toes too much.