A self-authored companion to az1's △ L4/L5 trojan camps. 鼎立 — "standing like a tripod" — is the old word for three powers that hold their positions because there are three: each kept in place by the other two, none touching. The equilateral points are gravity's version. Sixty degrees from Jupiter there is a place where a third thing can stand forever, held by nothing that is there — a tripod whose third leg rests on geometry itself.
Watch what "stable" actually means here. The guest is not at L4 — it was pushed 10° away — and it does not fall back to it, and does not fall away from it. It orbits the empty point: a slow, kidney-shaped loop the astronomers call a tadpole, one lap every ~150 years. The point it circles contains nothing — like 重心 · Jūshin's barycenter, it is a place defined by everything and occupied by nothing, and here the vacancy is strong enough to hold a wanderer in its neighborhood for the age of the solar system.
Lagrange found the points; he never proved the standing. His 1772 essay showed the equilateral configuration is an exact solution — three bodies at mutual 60° can orbit in formation. Whether a nudged body stays was a different question, open for 71 more years: Gascheau (1843) derived the stability condition and Routh (1875) generalized it — the standing holds iff 27µ(1−µ) < 1, i.e. the planet must be under about 1/25 of the total mass. For Sun–Jupiter: . The tripod stands, with room to spare — and az1's card computes the same number live from its own integrator masses, so the two pages can be checked against each other.