(sorry for bad English)A comment from person who actually handled medieval weapons:
The centaur would not all be archers. They will have as many archers as any human army, no more. Wars were not won by archers. They were won by foot soldiers, occupying and holding territory.
The height of centaur is a negative factor in close combat too. Centaur is restricted to attacking human from above only, while human can raise his shield above the head and attack centaur legs. In melee, centaur would have to keep human at greater distance than between humans.
Finally, the 50% of the army that day were pikemen. And at a first attempt against centaurs, the commander, unless very wise, would take even more pikemen.
With this in mind I conclude: the weapon of choice for centaur is a greatsword. Even more specifically flamberge. With long handle and extended ricasso. Greatswords were made specifically to fight pikemen in large groups.
I imagine centaur warrior: The armor is uneven. The front legs and the front horse part is armored best of all. Cuirass, may be, or at least very thick chainmail. Everything other is a light chainmail at best, otherwise it will be too heavy. Even more likely, it is thick cotton underlay with leather top. Cotton armor is very effective, but humans don't use it because it is so bulky. Centaur can. Anyway, it will be as much armor as they can carry without slowing down. Because speed is their weapon while size is their flaw.
He carries greatsword, maybe two axes and a heavy shield. At the beginning of the battle, they charge selected enemy units, keeping shields up to protect themselves from enemy arrows and bullets. When they reach few meters from enemy, they drop shields and grab greatswords with two hands. The length of centaur allows the creation of an effective holster for greatsword. If not, they just carry it on shoulder, holding it in one hand and shield with the other hand.
Centaurs use greatswords to break pikemen order. Pikes are useless in close combat. pikemen will drop pikes and draw swords, axes and clubs. While they do this, centaurs attack them with greatswords. When pikemen become axemen, the greatswords may be a disadvantage.(may be not) Then centaurs drop them and draw their axes.
Why axes? Because of their height, centaurs are at a disadvantage when fencing. As I already said, they have less angles of attack. For this reason, sword and shield are not as useful for them as axe and mace. They need weapons designed to break enemy defence, not to bypass it.
When centaurs of the first line start to fall, the warriors of the second line step in to keep pressure at the enemy.
Bows will not be used against enemy archers too. Why exchange blows when you can charge and trample? Bows will not be used against human cavalry too, the greatsword is very effective against knights. Bows will be used during initial phase of battle, during sieges and occasionally.
I doubt that centaurs will use kicks. While some legs are in the air, he is very unstable. A slight punch in the knee will topple him, and that is death.