If a person joins the congregation when the imaam is bowing, one of the following three scenarios applies:
1-He says takbeerat al-ihraam (saying “Allaahu akbar” to start the prayer) whilst standing, then he bows whilst the imaam is bowing. In this case he has caught up with the rak’ah with the imaam.
2-He says takbeerat al-ihraam whilst the imaam is bowing, but he bows after the imaam has stood up from rukoo’. In this case he is not counted as having caught up with the rak’ah with the imaam, and he has to make it up.
3-He bows straightaway without saying takbeerat al-ihraam. In this case his prayer is invalid, because he has omitted one of the pillars or essential parts of the prayer, which is takbeerat al-ihraam.
“The fuqaha’ are agreed that whoever catches up with the imaam in rukoo’ has caught up with the rak’ah, because the Prophet (peace and blessings of Allaah be upon him) said: ‘Whoever catches up with rukoo’ has caught up with the rak’ah.’” This was narrated by Abu Dawood, and classed as saheeh by al-Albaani in Irwa’ al-Ghaleel, 496; on p. 262 he said: one of the things that strengthens the hadeeth is the fact that the Sahaabah acted in accordance with this:
1 – Ibn Mas’ood, who said: “Whoever does not catch up with the imaam when he is bowing has not caught up with that rak’ah.” Its isnaad is saheeh.
2 – ‘Abd-Allaah ibn ‘Umar, who said: “If you come when the imaam is bowing, and you put your hands on you knees before the imaam stands up, then you have caught up (with that rak’ah).” Its isnaad is saheeh.
3 - Zayd ibn Thaabit, who used to say: “Whoever catches up with the rak’ah before the imaam raises his head has indeed caught up with the rak’ah.” Its isnaad is jayyid.
See al-Mawsoo’ah al-Fiqhiyyah al-Kuwaitiyyah, 23/133; al-Mughni, 1/298.