"There are places in the Solar System where the forces of gravity balance out perfectly. Places we can use to position satellites, space telescopes and even colonies to establish our exploration of the Solar System. These are the Lagrange Points."
From Fraser Cain's video on Lagrange points. A lot of pop sci Lagrange articles repeat and spread this bad meme. It just ain't so.
For the gravity vectors to cancel each other, they need to be equal and pointing in opposite directions.
The only L-Point where the gravity vectors pull in opposite directions is L1. And here the central body (Pluto) pulls harder than Charon. These two gravities don't balance out.
L3 and L2
Zooming in on the L3 and L2 points, we can see both bodies pull the same direction. These don't balance.
L4 and L5
Zooming in on the L4 and L5 points. Pluto pulls much harder. The angle between these vectors is 60º
The So-Called Centrifugal Force
There is a third player in these Lagrange tug of wars. What we used to call centrifugal force. This is not truly a force but rather inertia in a rotating frame. Here is an XKCD cartoon on this so called force:
Indeed, in a rotating frame, inertia sure feels like a force. The pseudo acceleration can be described as ω2r where ω is angular velocity in radians per time and r is distance from center of rotation. The vector points away from the center of rotation.
Putting Gravity and Centrifugal Force Together
Here's the same diagram but with centrifugal force thrown in (the blue vectors). Also the foot of the Charon gravity vectors are placed on the head of the Pluto gravity vectors -- this is a visual way to carry out vector addition.
For L1, Charon and Centrifugal Force are on the same team and they perfectly balance Pluto's gravity.
For both L2 and L3, Pluto and Charon are on the same team and they neutralize their opponent Centrifugal Force.
But what about L4 and L5? An observant reader may notice that the centrifugal force vector doesn't point away from Pluto's center. Adding Charon's tug to Pluto's tug moves the direction to the side a little bit.
Now the centrifugal force vector points from the barycenter. This is the common point of rotation around which both Pluto and Charon rotate. The same applies to L5.
L4, Charon's center and Pluto's center form an equilateral triangle.
The barycenter lies on the corner of a non-equilateral triangle.
And so it is with all the orbiting systems in our neighborhood. It is a 3 way way tug-of-war between centrifugal force, gravity of the orbiting body and gravity of the central body. Sometimes two players are on the same team, other places they switch. In L4 and L5 everyone pulls in a different direction. But in all 5 Lagrange points, the sum of the three accelerations is zero.