Question: When did the Ottomans conquer the Byzantine Emprire?
About the article "Turkey", section "The Seljuks and the Ottoman Empire".

Candidate passages:
  1304-1307: Successful war against the Byzantine Empire.
1322-1324: War against the Byzantine Empire.
1330: War against the Serbs, defeat in the battle of Velbazhd  1330-1332: Successful war against the Byzantines, victory at Rusokastro.
1341-1344: Successful war against the Byzantines.
1352-1354: First Ottoman raids.
1354-1364: The Ottomans conquer Thrace.
1365-1369: Hungarian occupation of Vidin, the Hungarians are eventually repulsed.
1366: Campaign of Count Amadeo of Savoy against the Bulgarian Black Sea coast.
1371-1373: War against the Ottomans.
1384-1396: War against Wallachia.
1385: The Ottomans conquer Sofia.
1388: Ottoman campaign in north-eastern Bulgaria.
1393: Fall of Tarnovo.
1396: Fall of Vidin, end of the Second Empire.  
In the early 15th century, the Peloponnese peninsula was divided between three powers: the Latin Principality of Achaea in the north and west, the Byzantine Greek Despotate of the Morea in the south and east, and Argos and Nauplia, Coron and Modon and some attendant forts, held by the Republic of Venice.
The Byzantines were actively trying to conquer the ailing principality, all the while the menace of the ever-expanding Ottoman Empire threatened all of them.
Carlo I Tocco, the ruler of the County palatine of Cephalonia and Zakynthos, of Lefkada and of the Despotate of Epirus, took advantage of the Byzantine-Achaean struggles to extend his domains into the Peloponnese: in 1407–1408 his brother Leonardo seized and plundered the fortress of Glarentza, in the northwestern Peloponnese, and in 1421 Carlo bought permanent possession of it from Oliverio Franco, who had seized it from the Achaean prince Centurione II Zaccaria three years earlier.
In February 1423, a shaky truce was brokered between Zaccaria, Tocco and the Byzantines by the Venetians, who were eager to establish a common front against the Ottomans, but this did not prevent a major Ottoman raid into the peninsula by Turahan Bey in summer 1423, nor did it stop the aggressive Byzantine despot, Theodore II Palaiologos, from raiding Venetian territory and even capturing Centurione Zaccaria in June 1424.
The Byzantines initially seemed content to let Tocco alone, as he too had open scores with Zaccaria, but war between the two powers was provoked in late 1426, when Tocco's forces seized the animals of Albanian herders during the latter's annual migration from the Byzantine-controlled central uplands to the plain of Elis. 
Andronicus III's reign was to be marked by Byzantium's last genuine and promising attempt at restoring "the glory that was once Rome".
In 1329, Byzantine troops were sent to meet the Ottoman forces who had been blockading, and in effect laying siege to, Nicaea since 1301.
Byzantine counter-attacks coupled with the scale of Nicaea's defenses had frustrated the Ottomans' attempts at taking any cities.
The fate of Nicaea was sealed when the Byzantine relief army was defeated at Pelekanos on 10 June 1329.
In 1331, Nicaea surrendered, resulting in a massive blow considering that it was the capital of the Empire 70 years prior.
Once again the Byzantines' military power was depleted and Andronicus III was forced into diplomacy as his grandfather was before him; in return for the safety of the remaining Byzantine settlements in Asia Minor, tribute would be paid to the Ottomans.
Unfortunately for the Byzantine Empire, this did not stop the Ottomans from laying siege to Nicomedia in 1333; the city finally fell in 1337.
Despite these setbacks, Andronicus III was able to score a few successes against his opponents in Greece and Asia Minor; Epirus along with Thessalonika were subjugated.
In 1329, the Byzantines captured Chios and, in 1335, secured Lesbos.
Nonetheless, these isolated Islands were isolated exceptions to the general trend of increasing Ottoman conquests.
Furthermore, none of the Islands were a part of the Ottoman domain; their capture demonstrates the potential that the Byzantines had at the time of Andronicus III.
Byzantine military ability would be further weakened by Serbian expansions into recent acquisitions by Andronicus III (Epirus) and finally by a devastating civil war that would subjugate the Byzantine Empire as a vassal to the Ottomans. 
Following the Byzantine defeat at Nicaea in 1331, the loss of Nicomedia was only a matter of time for the Byzantines.
The Byzantine emperor attempted to bribe the Ottoman leader Orkhan, but in 1337, Nicomedia was attacked and fell to the Ottomans.
The Byzantine empire did not recover from this defeat; the last Anatolian stronghold of Byzantium had fallen, except Philadelphia, was surrounded by the Germiyanids till 1396.  
Ottomans wanted to destroy Solomon I and his supporters and to finally conquer western Georgia.
The aim of Georgians was to clear Imereti from Turks.