The French invented the “menage a trois,” so leave it to them to conjure up a particularly messy one that looks at love from “Both Sides of the Blade.”

Titled “Avec amour et acharnement” in French, and first-titled “Fire” in English by director Claire Denis, this Juliette Binoche drama has heat and hurt and a whiff of intrigue as Denis (“High Life,” “Beau Travail”) peels away layers of background that reveals more and more of the true nature of her characters.

For her third film with Denis (“Let the Sunshine In” was the first), Binoche gives us Sara, a smitten but wholly adult 50something who is all over her lover of several years, Jean (Vincent Lindon of “Titane”) — in a solitary romantic cove on the Riviera, in their apartment — filling his ear with “Mon amor, mon amor, mon amor” as they make love.

But a glimpse of a former lover, a younger man, François (Grégoire Colin), rattles her and sends Sara into reveries of the past.

“François, François, François,” she purrs, hoping no one hears. Being a radio talk show hostess, she hides nothing from Jean, who is 60ish and struggling to keep a handle on his acting-out teenaged son (Issa Perica), who lives with Jean’s constantly-calling elderly mother (Bulle Ogier), who is raising him.

It turns out Jean and Sara’s meeting is something she remembers more fondly than him, a meeting that happened when he was still married and she was still with “François, François, François,” who left her alone at the end of the night while Jean showed concern for her over her then-lover’s callousness.

“Why am I with the one who leaves?” she asked herself, recollecting that night to Jean.

So much is revealed in that simple exchange — Sara’s willingness and ability to get the she wants (casting Binoche makes that a no-brainer), her tactless confidence in mentioning that she’s seen François, and Jean’s solicitous devotion, a devotion that can be re-directed.

When François reaches out to Jean with a business proposition, we figure the younger man out with his first line (in French, with English subtitles).

“So, you kept her?”

It takes Jean and us a moment to figure out he’s talking about Jean’s 35 year-old Mercedes. And then we and he wonder if that confusion is intentional, what François really wants and what he and Sara might be up to, or get up to if given the chance.

“Both Sides of the Blade” is a film of chats that sound like interrogations, big revelations that drop in normal, everyday transactions and mixed emotions all around. Jean’s acceptance of a business offer from François might seem easy and natural, but Lindon’s tight-lipped playing of it suggests just how fraught all of this whole situation is to him.

Is Sara adult enough to resist swooning? How dark is Jean’s dark side, and when will he show it? And just what is François on about with this “offer?”

Denis, who co-wrote the script with Christine Angot, gets at the fragility of relationships and the narcissism of love in a film where the stakes might be low, but promise to go higher at any moment.

Every character has her or his “sketchy” side, impure motives or selfish desires. Even though we’ve glimpsed an idyllic romance in the opening scenes, the three corners of this menage seem co-dependent. Perhaps that loving couple needed whatever drama in the past launched and fed that relationship, and everything they do now is overcompensating for that.

But Denis won’t let us wrestle with that directly as we listen in on Sara’s radio show or lose ourselves in whatever drama’s going on with Jean’s biracial son Marcus. Thus, the testy and tender resolution simply sneaks up on it.

Life and love on can be harrowing and messy. And maybe you have to be your most selfish to get what you want from it, grabbing that knife from “Both Sides of the Blade.”

Rating: unrated, sex, nudity, profanity

Cast: Juliette Binoche, Vincent Lindon, Grégoire Colin, Bulle Ogier and Issa Perica

Credits: Directed by Claire Denis, scripted by Christine Angot and Claire Denis. An IFC release.

Running time: 1:56