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Ancient Macedonians
Was Alexander the Great Greek? Well, most scholars that have been lively in contact with the heritage of Alexander and his Empire would deny that fact. They are explicit on the view that Alexander was Ancient Macedonian (the adjective ancient being added to make a distinction with present-day Macedonians, that are mainly Slavic).
Ancient Macedonia consisted of present-day Aegean Macedonia (part of the Greek Macedonia), the southern part of present-day Republic of Macedonia, and a small portion of the Bulgarian part of Macedonia. This means that historical Macedonia is another term, and Republic Macedonia is also another term. Completely confusing, I know, but just continue reading without prejudices and will get to the point where I'll make the distinction between all of them. However, the largest part of the territory of Ancient Macedonia is in fact the Greek part of Macedonia. This is all, strictly geographically speaking.

Ancient Macedonia
Macedonians came to prominence as a key-player in the Balkans somewhere around the 7-6th century BC with the establishment and strengthening of the Macedonian Kingdom (it's first ruler being Perdicas I). He and the long line of his successors, menaged to strengthen and expand the kingdom to the neighbouring lands of the various Balkan tribes (Ilirians, Thracians, Dardanians, Paionians, Greeks etc.)
When talking about the ethnic origin of Macedonians, one has to know that Macedonians didn't have a written language of their own and most of what we know comes from the Greek sources that were written by Greek historians. From archeological evidence, we can say that there are many artifacts from the Archaic period of the Ancient Macedonians (from the end of prehistory to the contact with the Persians) that connect them to the other Balkan peoples, and very little to the Greeks. In addition it is considered that Macedonians' origin can be traced to the Brigian tribe (an Indo-European people that could have migrated from Asia Minor), just like the origin of other Balkan tribes, but the Greeks.
If one also looks in the religion of the Macedonians as reference there are also facts that undoubtly show their distinctivness from the Greeks. (the spelling is probably false

The same can be said about the language. Ancient Macedonian language was spoken, but never written. Today there are only about 100 words ("glossae") from that old language from some stone engravings (with the Greek alphabet of course), and scientists still work on their encryption. Most have little in common with the Greek language however.
Another ethnographic element would be the customs and symbols which were many and unique. You already know the Sun with sexteen rays of Vergina that used to be on Macedonian army shields, then the lion as the symbol of royal power (did you know lions existed in the Balkans that time?), the custom of confesion, the custom when the army would cleanse itself by passing between the two halves of a cut dog (ick! I know

All in all, one is basically right when they say Ancient Macedonians were a "barbarian" tribe, that had little in common with the Greeks.
Macedonians were not really allowed in the Olympic Games. There was a case when the Macedonian King Alexander I, wasn't allowed to participate, but he proved his lineage to the city of Argos Orestikos (a Greek city in Macedonia), and in the end he was allowed to participate (because after all he was the king) and won. And hence, why there were Games organized in Dion by Macedonians.
Alexander I was also known as Alexander I Phillhellen (meaning "loves Greeks"). If he was considered a Greek, they wouldn't have called him Greek-loving, don't you think? He ruled in a time of great hellenization of Macedonian culture (more in the following paragraph).
All this said so far, is absolutely true about the Archaic period of Macedonian history (up until the Greek-Persian War, V century BC), after which the Classic and Hellenistic periods took place.
With the increasing contact with the Greek colonies on the Macedonian coast and the strong, amazing and influental Greek culture (which I personally admire btw), came the more and more massive hellenization (becoming Greek) of the authentic Macedonian culture. Greece was a centre of the world at the time and it's understandibly enough why the Macedonian Kings wanted to mingle in Greek affairs. As much as the Macedonian Kings conquered many of the neighbouring Balkan tribes, they could never truly call themselves superior without conquering the Greek city-states too. They became increasingly dependant on Greek well-manufactured imported goods, culture and recognition. The Macedonian Kings transfered the capital from the mountainious city of Aegaea to the more coastal Pela. They also brought famous Greek thinkers to the court, the most notable being Aristotle, brought by Phillip II to tutor his son Alexander (later the Great). Alexander's most beloved book was Homer's classic "Illiad". All Macedonian aristocracy could speak Greek, and probably the commoners too. The Greek language introduced new words for many things the poorly-educated and technologically backwarded Macedonians didn't have names. Even the names of the dieties became Hellenized, just as it happened with the dieties of the other Balakn people. This is the time when temples and other Greek-exlusive culture marks began to take hold in Macedonia. Before (during the Archaic period) there was no such culture or Greek architecture in Macedoania. The Classical period is the epoch from which the biggest amount of sources comes from, and that's why it was long ago thought that Macedonians were Greeks, i.e. of Dorian or Achaean origin. (In fact it's a matter of policy as well as misinterpretation)
An evidence of the Greek resentment towards Macedonian rule is the hatred with which Athenian statesmen spoke of Phillip and his intentions to conquer /unite (as pro-Macedonian parties in the Athenian assembly spoke) Greece, in order to fight against Persia. Demosthenes was the most prominent anti-Macedonian and Philip II hater and one of his most famous work is the Philipics, a series of speeches aimed to attack Phillip and his policy as well as nationality and personality. In one of his speeches he says: "The Macedonian King has nothing in common with Hellada (Greece), nor with her culture. He is barbaros (meaning "alien, non Greek"), despot and tyrant that won't save Greece." He organized an alliance against Philip, but that didn't save the Athenians, because with the battle of Chaeronea (338 BC) all city-states, but Sparta fell under Macedonian rule. The following year, Greece was united under the patronship of Macedonia and Philip declared war on Persia. However, soon he was killed by some Macedonian aristocrats, and Alexander was appointed king by the army. After he consolidated his position, both interanal with the generals, with the naighbouring Balkan tribes and the Greek city-states as well, he set to conquer Persia itself, and soon thereafter the world. And then is blah, blah, blah... you know the rest.
(It should also be noted that when Philip was killed the Athenians were overjoyed. They offered a sacrifice to Athena and praised the men who killed him.)
The next period is the Hellenistic period (from the death of Alexander, 323 BC, to the coming of the Romans). This wasn't a uniquely Macedonian period, but a global one. It was characterized with mixing the Greek culture (and to a small extent the hellenized Macedonian culture) with the Middle Eastern cultures. A new culture was created, called Hellenistic, with a strong Greek cultural and linguistic influence. It was then when many MIddle Eastern cults spread to Greece and Macedonia and were in a way re-introduced in Macedonia, who had already much forgotten her Brigian origins. However, this was not a rediscovering of some kind, but an introduction to a whole new culture - the Hellenistic. By this time, art and crafts were widespread in Macedonia, and they were largly influenced by the Hellinistic Age, just as in other parts of Alexander's Empire.
Still, the Macedonians in Alexander's army remained truthful to their mother language, and it can often be heard from ancient sources that often the Macedonian generals used Macedonian in their internal communication, as well as that Alexandar often spoke Macedonian when he lost his temper. This undoubtedly prooving that Macedonian was not the same language as Greek, since it was so obvious to make a distinction and seperation between the two.
The Hellenistic period historically was characterised by the wars of the Diodochs (the Seleucids, Ptolemids, Cassandrids, Antigonids, Lysimachids, Bactrians, Atropatinids, etc.); Carthage; the emergence of Rome, Kushan and Parthia; and the last stand of the ancient Greeks (the Achaean and Aetolian Leagues struggle to remain free of Rome and the Diodochs.
The Diodochs were the Macedonian generals from Alexander's army and powerful members of the Macedonian aristocracy. They all wanted to take Alexander's place, but instead they only menaged to seperate the empire into seperate pieces. The most important were Egypt (ruled by the Ptolomeids; that would mean Cleopatra was Macedonian), Syria, Mesopotamia, Phoenicia, Asia Minor all ruled by the Seleucids, and the region of Macedonia and Greece ruled by the Antigonids.
Needless to say, instead of being grateful for all that Alexander did for their culture, the Greeks were happy when Alexander died and they immediately rebelled against the Macedonian rule (more precisely Antipater, the chief general of Alexander; not an Antigonid, that dinasty was formed later.). The ensuing Lamian War (a liberation war for the united greek city-states) ended with them being defeated. However, later things got messy and they gained their independence. Their following leages and alliances were aimed against the Macedonian kings, and in this respect did the two leages (Achaean and Aetolian) functioned, i.e. allied with Rome to fight the Macedonians. But the Roman moto: "Divide et impera." prooved itself one more time, and they were soon conquered by the Roman army as well as Macedonia. The last Macedonian King to fight the Romans in the Roman-Macedonian Wars (3-2rd century BC) was Philip V and after that Macedonia was turned into a Roman province.
That was all on Ancient history. No politics in here. From all this we can conclude that Alexander's origin is debatable to the very least (and most facts obviously point to him being non-Greek).
However, the latter part is a terrible mess and it would be hard for me to explain it without writing a whole novel, so I'll try and add only the info necessary for you to understand the general idea.