This second aorist -- here in the first person singular or third person plural; either way, evidently quoted from somewhere -- is assigned in modern grammars and dictionaries to tugxa/nw (tau 1147, web address 1), but it belonged originally to the same paradigm as teu/xw and remained so in the paradigm known to the Suda (see tau 435 at length, also on the source of this group of entries). In Homer’s Odyssey, the warriors from Troy look for those who can "provide" their return (19.314) in order that they may "achieve" return (6.290). It is used most commonly, in the epics, of successfully hitting the mark with an aimed spear, arrow or other missile; together with its antonym from a(marta/nw, it is used in a phrase that became proverbial for failure, h)/mbrotes ou)d’ e)/tuxes, "you missed and did not hit", i.e. despite your preparations (Iliad 5.287; see eta 286, and cf. the unerring spear of Procris at pi 2484). Secondly, it is used in senses either where destiny or chance plays a role: (Iliad 8.430) or of happening to be at a particular spot at a particular moment (not necessarily by design but rather by coincidence or chance), see Iliad 11.116, Odyssey 12.106, and cf. the metrical alternative e)tu/xhsa at Odysssey 14.334 = 19.291, 21.13, the last in the sense of meeting a man. It has a third use in Homer, of the thing that met me, came my way by luck (Iliad 11.684, cf. tau 1147).
In Pindar tyche becomes a concept of succeeding under the fashioning power of the gods (or a trainer), almost visually of arriving at, or “hitting”, the finish line in a race (cf. Villard Leglay's excellent section on Pindar, 109ff.). This visual image, together with that of the moment when a spear hits its victim, may remain constant in all later uses.
In an important note on Thucydides' use of the verb for the arrival of Demosthenes and the Messenians at Pylos during the Peloponnesian War, Gomme observed that the historian regarded being at the right place at the right time as a goal of ancient military tactics that depended not a little on chance: “tugxa/nein does not necessarily mean that an event was accidental, but that it was contemporaneous… that the Messenians had arrived by arrangement with Demosthenes is obvious, and is implied by 3.3; but to arrive at exactly the expected time was to some extent fortuitous and fortunate. It was similarly fortunate for the Athenians, and again not planned by them, that the Spartans had not blocked the entrances; but it was not chance." At the height of Athenian trust in the power of human reason and foresight after the Persian Wars, tyche became an explanation of the unforeseen and unforeseeable events to which human reason must adjust. Its place in military strategy remains a commonplace in Polybius and Plutarch (tau 1234, cf. alpha 537, alpha 2310, epsilon 695 and cross-references) and was immortalised in Caesar's "Let the die be cast" (epsilon 3013).
The conception of tyche in general, but especially in the historians Thucydides and Polybius, has given rise to a small publishing industry (see bibliography at tau 1234). To the degree that this abstract noun developed in Greek philosophy (e.g. the lost treatise of Demetrius of Phalerum) and popular thought, from the 4th. Century B.C. on, the meaning of pure chance and the trappings of a goddess, so this aorist, especially in its third-person impersonal use, took on the overtone of "it happened through meaningless chance." See also the idiom w(s e)/tuxon/ w(s e)/tuxen, ("at random") at alpha 463, alpha 1193, alpha 2108, alpha 2220, alpha 4308, alpha 4538, epsilon 571, epsilon 2406, epsiloniota 66, epsiloniota 77, iota 172, kappa 1272, mu 768, omicron 804, omicron 805, omicron 975, sigma 974, upsilon 528.