煮える and 煮る: ingredients cooked in liquid and the process
Distinguish cooking ingredients in liquid from their cooked state, and compare boiling liquid with ingredients becoming cooked.
Cook in liquid
鍋で野菜を煮ています。 says that someone is cooking vegetables in liquid in a pot. 煮る, read にる, takes を for the ingredients and has the te-form 煮て.
弱火で10分煮てください。 asks for ten minutes of cooking over low heat. 弱火で gives the heat level and 10分 the duration; the negative form of 煮る is 煮ない.
The ingredients become cooked
じゃがいもが煮えました。 says that the potatoes have cooked in liquid. 煮える, read にえる, makes the cooked ingredient the subject and has the te-form 煮えて.
まだ中まで煮えていません。 means that it is not yet cooked through. A hot surface does not establish that the whole ingredient is cooked enough.
Boiling liquid and cooking ingredients
お湯は沸いていますが、野菜はまだ煮えていません。 says that the water is boiling but the vegetables are not yet cooked. The liquid boiling and the ingredients being ready need not occur at the same time.
煮た野菜を皿に盛りました。 says that someone served the vegetables cooked in liquid on a plate. 煮た modifies the noun with its cooking method and differs in meaning and spelling from 似た, meaning similar.
If water is 沸いている, must the potatoes also be 煮えている?
Show the answer and why
No. The potatoes may need more time to cook through even while the water is boiling. Check the liquid's state separately from the ingredients' readiness.