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[This website](https://nihongokichijoji.blog.ss-blog.jp/2018-02-10) explains
that:

> 「～ところへ／に」は、ある段階における状況を変化させるような出来事が起こることを表すと説明した方がいい。  
>  to indicate event that will change the situation in certain stage  
>  (this is my own translation (it might be incorrect))

> 「～ところで」は、前の動作が終わり、一区切りがついた時点で、後の変化が起きる、又は動作を起こす。  
>  In a point of time when the action is over and there's a break, another
> action happens / perform another action  
>  (this is my own translation too)

> ふっとうしたところ **へ** とりにくを入れます。

I think ところで is more suitable (even though I think it's wrong) in this
sentence because 「前の動作が終わり、一区切りがついた時点で、後の変化が起きる、又は動作を起こす。」

前の動作が終わり = ふっとうした 後の変化が起きる = 肉を入れます

But anyway, I feel like I don't understand the difference between ところで and
ところに/へ at all because when I feel it's unnatural when I use it. See also
[Usage of ところ to represent time with に and
で](https://japanese.stackexchange.com/questions/66698/usage-
of-%e3%81%a8%e3%81%93%e3%82%8d-to-represent-time-with-%e3%81%ab-and-%e3%81%a7)
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question
grammar particles に-and-で
What is the difference between ところに/へ and ところで?
A simple explanation about the difference between ところに/へ and ところで is that the
former takes not only time but space aspect of an event into consideration.

  * _X_ ところで _Y_ : _Y_ at the same moment _X_ happens
  * _X_ ところへ/に _Y_ : _Y_ at the same moment, **same place** _X_ happens

> ふっとうしたところ（へ／で）とりにくを入れます

Both will work in this example, because according to my common sense, it
instructs you to put chicken into the pot when water in it boils. However,

> ○ ふっとうしたところ **で** とりにくをやきはじめます  
>  × ふっとうしたところ **へ** とりにくをやきはじめます

because, if I'm correct, you can't roast meat in the pot with boiling water
inside. In order to do so, you have to bring another utensil to put your meat
in, and it is not the **same place** the boiling takes place in.

A little deeper reasoning for that is that へ/に are particles to mark the
destination of action (i.e. towards it), unlike で which denotes general
background settings. Even they are idioms, their usages are bound by grammar.
That also means you can't use ところへ/に when the verb don't accept it.

> ○ 渋谷を歩いているところに先生が来た  
>  × 渋谷を歩いているところに先生と会った  
>  ○ 先生が渋谷を歩いているところに会った

When you use 会う "meet / run into" with the other one marked by と, you can no
more use に in this construction.
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answer