What does Buddhism say about dealing with negative people?

What the Buddha said about this was already answered in the last question about The Simile of the Saw:

"Suppose that a man were to come along carrying a burning grass torch and saying, 'With this burning grass torch I will heat up the river Ganges and make it boil.' Now, what do you think — would he, with that burning grass torch, heat up the river Ganges and make it boil?"

"No, lord. Why is that? Because the river Ganges is deep & enormous. It's not easy to heat it up and make it boil with a burning grass torch. The man would reap only a share of weariness & disappointment."

"In the same way, monks, there are these five aspects of speech by which others may address you: timely or untimely, true or false, affectionate or harsh, beneficial or unbeneficial, with a mind of good-will or with inner hate. Others may address you in a timely way or an untimely way. They may address you with what is true or what is false. They may address you in an affectionate way or a harsh way. They may address you in a beneficial way or an unbeneficial way. They may address you with a mind of good-will or with inner hate. In any event, you should train yourselves: 'Our minds will be unaffected and we will say no evil words. We will remain sympathetic to that person's welfare, with a mind of good will, and with no inner hate. We will keep pervading him with an awareness imbued with good will and, beginning with him, we will keep pervading the all-encompassing world with an awareness imbued with good will equal to the river Ganges — abundant, expansive, immeasurable, free from hostility, free from ill will.' That's how you should train yourselves. [...]

"Monks, even if bandits were to carve you up savagely, limb by limb, with a two-handled saw, he among you who let his heart get angered even at that would not be doing my bidding. Even then you should train yourselves: 'Our minds will be unaffected and we will say no evil words. We will remain sympathetic, with a mind of good will, and with no inner hate. We will keep pervading these people with an awareness imbued with good will and, beginning with them, we will keep pervading the all-encompassing world with an awareness imbued with good will — abundant, expansive, immeasurable, free from hostility, free from ill will.' That's how you should train yourselves.

Of course it isn't a practical to say 'Just have loving kindness towards that person all the time'. But having in mind this simile can be great for such situations. One way of dealing with these situations is morality, having this attitude 'I am angry, but I'm not gonna let my speech & action be affected by that'. In some extreme cases you might consider leaving the situation, knowing 'I am not up to this yet'. The best way to do loving-kindness for this person is when this person is absent or you're in a different place (But don't begin with this person, the difficult persons have to come at the end of metta meditation after the loved ones). From a Buddhist point of view being confronted with difficult people is a great teacher. It shows you where there is attachment and work to do. Don't think 'He is stealing my peace', nobody can make you react to things except yourself.

Also from MN 8

But herein, Cunda, effacement should be practiced by you:

1. Others will be harmful; we shall not be harmful here — thus effacement can be done.
2. Others will kill living beings; we shall abstain from killing living beings here — thus effacement can be done.
3. Others will take what is not given; we shall abstain from taking what is not given here — thus effacement can be done.
4. Others will be unchaste; we shall be chaste here — thus effacement can be done.
5. Others will speak falsehood; we shall abstain from false speech here — thus effacement can be done.
6. Others will speak maliciously; we shall abstain from malicious speech here — thus effacement can be done.
7. Others will speak harshly; we shall abstain from harsh speech here — thus effacement can be done.
8. Others will gossip; we shall abstain from gossip here — thus effacement can be done.
9. Others will be covetous; we shall not be covetous here — thus effacement can be done.
10. Others will have thoughts of ill will; we shall not have thoughts of ill will here — thus effacement can be done.
11. Others will have wrong views; we shall have right view here — thus effacement can be done.
12. Others will have wrong intention; we shall have right intention here — thus effacement can be done.
13. Others will use wrong speech; we shall use right speech here — thus effacement can be done.
14. Others will commit wrong actions; we shall do right actions here — thus effacement can be done.
15. Others will have wrong livelihood; we shall have right livelihood here — thus effacement can be done.

the fact that you (and all of us) are susceptible to other people's opinions is due to our ego, the sense of precious self

in this respect development of understanding of anatta is a useful strategy

and an excellent advice is given in the Ā­ghātapa­ṭi­vinaya­ sutta 2 (AN 5.162) of only attending to pure qualities of a person

With an ensō (and "the beauty of imperfection") as an avatar, you might like a Zen story: Obedience. The story reminds me of my mother, a nursery school teacher (very young children) -- she said that instead of trying to control their behaviour by criticizing them, "I wait until they do something right and then praise them" (e.g. "Thank you for putting your work away", or "Thank you for sharing", or etc).

One of the funniest stories I remember was of one of her assistants who, instead of directly addressing and criticizing the whole class, raised her voice just loudly enough for everyone to hear and said to my mum: "Mrs. Teacher, where have all the quiet children gone?"

I can't easily fathom your relative's motive. You're talking about "self image": could it be that he wants you to be like him (or, like his image of himself), or that he wants to be like you?

Would it be fair to say that you are already behaving well? You say that your peace is stolen and your equanimity robbed, but I'm hoping you don't fuel the flames, e.g. try to take revenge on him and so on. If you can stay untainted by misbehaviour, harsh speech and so on even when he's making "cold remarks" then you're already doing well. This article for example claims that, "The satisfaction of leading a blameless life is the highest form of satisfaction that a layman can have".

I just want to add that "blameless" doesn't mean "holier than thou", which would be more self-image.

Anyway, to answer your question, "I'm wondering what Buddha taught on this?"

I wonder whether you might be interested in reading about "fame". That might be another side of the coin of what you're talking about -- i.e. being motivated by how we think others see us (and maybe identity-view too, how we see ourselves).

This search for 'fame' and 'lay' on Access to Insight finds articles that may be a useful introduction, or another perspective.

One more thing, is that your language includes the feeling of being "robbed" of your equanimity.

I think that's kind of a dangerous view to have; for example the Dhammapada starts with,

3. "He abused me, he struck me, he overpowered me, he robbed me." Those who harbor such thoughts do not still their hatred.

Perhaps that should be contrasted with the Zen story, The Moon Cannot Be Stolen.

You wrote, "Sometimes I am mindful of it, and realize the attachment is there"; but attachment isn't only to "self image or sense of self-worth in life" -- an attachment can also be to a "feeling" (such as equanimity), so thinking of it as "my equanimity" might be dangerous.

And maybe you're treating equanimity like glass, as if an equanimity could be broken; instead maybe you could see it as more fluid, less rigid or resistant, perhaps more like water (an ocean) or air or fire.

Or maybe a traditional metaphor is of the moon reflected in water, if the water is troubled then the image is harder to see, but the moon itself is not troubled.

The Buddha taught on three levels:

1. morals - skilful action;

2. concentration - non-judging mind;

3. wisdom - not-self.

Each of the three levels of practise can be used to address such issues.

On the moral level, the Buddha taught members of a family should cultivate four qualities:

1. truthfulness; honesty; straightforwardness (sacca)

2. training in self-improvement (dama)

3. patience; forbearance (khanti)

4. generosity; sacrifice (caga).

Therefore, on the moral level, you may straightforwardly, honestly & patiently communicate to the relative that their behavior is not constructive to a good family relationship & that you believe they should improve their manner of relating towards you.

Families can be different to other social interactions in that families are a constant. Being a constant, we can be more firm in our expectations & hopes. If a relative behaves in a rude manner towards me, I straightforwardly tell them I am not interested in such interactions.

The Buddha taught families that do not practise the five precepts (which includes good speech) will end up with troubles. The Buddha compared family members as 'fires' that must be tended safely.