Participles

Murrine sees Dikaiopolis coming into the courtyard. This com-ing form is called Present Participle.
The word participle comes from Latin (pars, part and capere, to take) and means: to take part.
Participle forms participate in the action:
He runs away laughing.
His main action is running, but he also laughs, so laughing plays its part, participates in the on-going action and is therefore called a present participle.

Which is main action and which is participating only? It all depends on the given situation:
I watch TV and keep an eye on my new, not quite house-trained puppy.
Main action: watch TV
participating action: keep an eye on puppy.
So, using a present participle we would say:
I watch TV, keeping an eye on my new.......
I used to listen to the radio while I did my homework.
main action: did homework
participating action: listen to the radio
Using a present participle we'd say:
I used to do my homework listening to the radio.

We don't use plain participles much in this context, but prefer to circumscribe with "while" or "and". The Greeks on the other hand, and the Romans, made ample use of participle constructions.

I listen to the radio while doing my homework:
τὸ ἔργον παρασκευαζόμενος μουσικῆς ἀκούω.

Why present participle? Because the participating action was, is or will be taking place at the time of the main action. The emphasis is on at the time, whenever that time may be:
       Roaring terrifyingly, the Minotaur will attack Theseus.
       Roaring terrifyingly, the Minotaur attacks Theseus.
       Roaring terrifyingly, the Minotaur attacked him.
The attacking and the roaring taking place at the same time, the roaring form is present participle in all 3 sentences.

Present participle: middle forms

Declension of present participle forms

Since they replace full sentences, or at least parts of sentences, present participles have to agree with their subject.
a working man is ἀνὴρ ἐργαζόμενος 
a working woman γυνὴ ἐργαζομένη 
a working beast  θηρίον ἐργαζόμενον 

They are declined like 1st/2nd declension adjectives.
As the base form (masculine singular -όμενος) tells us, they are accented on the antepenult, unless the ultima is long:
All the girls are admiring the dances, so
πᾶσαι αἱ κόραι θαυμάζουσι θεώμεναι  (short ultima, accented antepenult)
ἡ Μέλιττα θαυμάζει θεωμένη (long ultima, accented penult)

We greet people arriving: χαίρομεν ...
.. τὸν πατέρα τὸν ἀφικνούμενον (short ultima, accented antepenultima)
.. τοὺς φίλους τοὺς ἀφικνουμένους (long ultima, accent shifts from antepenult to penult)