When there stand two things, we call they stand “in parallel,” what do you call three or more things stand “in triangle”?
Based on clarification in the comments, I believe that 並立 and 鼎立 can be reasonably translated to an established English phrase: "To stand as equals."
This is a phrase that can be ascribed to rivals, colleagues, enemies etc. and can contrast with "standing above" or "standing below" in which height or vertical position is a metaphor for status.
For example, one might say that at the outset of the Democratic Primary, Hillary Clinton appeared to stand above Bernie Sanders, but he has since proven that he can stand as an equal with her.
Some meaning is lost in that 鼎立 seems to refer to two peers who stand above all others, but in context it is often implied that this is the case. And of course "standing as equals" does not specify a number, though it is most often referring few parties.
After some more thought, I believe trilateral will work for your context of three (and multilateral for more than three).
: involving three groups or countries
Merriam-Webster
The analog for two parties would be bilateral.
When something is bilateral it has two sides or it affects both sides of something. Discussions between two political parties are called bilateral because both sides get to share their views.
Vocabulary.com
Consider the following article describing an agreement signed by China, Japan, and South Korea (bold text is mine):
On 13 May 2012, China, Japan and South Korea signed the Agreement among the Government of Japan, the Government of the Republic of Korea and the Government of the People’s Republic of China for the Promotion, Facilitation and Protection of Investment (“Trilateral Investment Agreement“). The Trilateral Investment Agreement is the first legal framework between the three East Asian nations regarding investment and once in force, aims to enhance and protect investments made trilaterally, whilst also paving the way for a potential Free Trade Agreement between China, Japan and Korea. ...
In addition to the more common protections that are covered under bilateral investment treaties already in force between China, Japan and South Korea (“Existing BITs“), such as fair and equitable treatment, most favoured nation treatment and protection against expropriation, the Trilateral Investment Agreement also promises improved government transparency, express protections for intellectual property rights and exceptions that will allow governments of the host State to take prudential measures to ensure the stability of their financial systems. It also identifies international arbitration as the key dispute resolution mechanism for foreign investors.
Herbert Smith Freehills
Edit: After reading the comment conversation I now think the first shot definition was way off.
Though, I believe even more now that "Tripartite" is a correct answer. It means three parties involved in the same thing. Parties here can be both individuals, groups, countries or any number of things. It can then be combined with several other words to fine tune the meaning, such as "Alliance" to mean three groups working on the same side or for the same ideal (in parallel/along the same lines). Further, it can also be combined with words like "Dispute" to mean three groups working against one another, that is, "contesting" (which could harken back to the three-cornered contest that I and others have seen).
For three people facing each other, I would say that it sounds almost metaphorical. Imagine three people standing in a triangle. In order for all of them to face each other they cannot be looking in the same direction as anyone else. This could mean "Because they are not looking in the same direction, they are not focusing on the same ideals and therefore working against one another." Which might mean the word's default is the "in contest" one, but can be altered with other Japanese words or phrases to mean "three working together" or "three in parallel."
Another thing I thought of is also a bit metaphorical, but changes the way I'm using parallel. Parallel lines never touch, by definition and so it would sometimes make sense that they aren't working together, but you can also look at them as trying to get to the same point. Racing to the same point and not working together can be seen as contesting against each other for the same thing. Parallel in this metaphor could mean what he's looking for, but not a standard way to look at the word in English.
But that's still a bit fuzzy. I'll link definitions later if this is still holding up. On my phone and it's tricky to do.
Two parallel lines never cross or intersect with each other and the (minimum) distance between them remains the same to infinity.
(Please note that I emphasize the “and” above solely to address the “binarity” of parallelism, for although “parallel” may not be binary in the sense that an infinite number of lines can be drawn that never intersect with each other, the constant equidistance part of the description of parallel lines can only be satisfied by a maximum of four parallel lines in a three dimensional Cartesian coordinate system and in the grand scheme of things, 4 seems closer to being binary than to being infinite.)
(link to a question on ‘Quora’ about the maximum number of points (and therefore parallel lines?) that can be equidistant from each other and one of its answers that says that the maximum is 4)
Contrary to parallel lines are the ones that eventually cross or intersect with each other and a tripod is formed when three non-parallel lines intersect/cross/come into contact at the same point, therefore I think “in intersection” (in the sense of “coming into/having contact with”) would convey the opposite of “in parallel” in the literal sense, but in a figurative sense, I would equate “in parallel” to “[standing] safely/comfortably apart” and its opposite (in tripod) to “[standing] dangerously close or too close for comfort”
For the sake of world peace, two opposing forces (world powers, for example) are perhaps best kept “safely/comfortably apart” from each other, running/standing [in] parallel with the [equi]distance between them maintained by, among other things, the notion of “balance of power.”
(To the extent that “being/working in parallel” might [or even has to] imply to some that the two forces have the same interests and goal, one could argue that they both have the same interests, i.e., self interest/preservation and the same goal, i.e., peaceful coexistence.)
On the other hand, the relationship between entities (such as the EU powers mentioned in the question) whose paths purposely cross and intersect on a regular basis, although giving rise to great opportunities to resolve Europe’s problems and achieve its unity, also gives rise to more opportunities for head-on confrontations and such a close relationship could be characterized as one that’s “dangerously close” or “too close for comfort,” perhaps similar to the feeling some participants experience in a three-legged race, especially when a virtual stranger has been assigned to be their partner.
The terms vie, mexican stand-off and bilateral/trilateral/multilateral have already been proposed or used in the original post.
You have also used the term "power balance", but I would like to suggest the more formal balance of power as an English term that captures some of what I understand of your descriptions of the Japanese terms.
Balance of power 1 A situation in which states of the world have roughly equal power. 'As already noted, Soviet power was certainly an important element in the Cold War balance of power in East Asia.' - ODO
(Note: there is a second way this term term is used: a small party that can tip the balance to one side or the other may be said to hold the balance of power, but this is not the sense I'm discussing in this post.)
Wikipedia quotes Kegley & Wittkopf:
The balance of power theory in international relations suggests that national security is enhanced when military capability is distributed so that no one state is strong enough to dominate all others.
Here is an example of the usage of the term:
First, the general balance of power serves to prevent the system of states from being transformed by conquest into a universal empire ....
- Bull (1977, pp.116-117) as quoted by Niou and Ordeshook in "The Balance of Power: Stability in International Systems (p.82) as Bull's summary of the meaning of balance of power
The phrase includes the notion that the parties (states, in the context of international relations) to this balance are of equivalent power. For example, if the three Chinese states you mentioned in comments were of equivalent strength, each of the three states would be an element of the balance of power in that context. The phrase also has something of a nuance of facing each other, though they don’t always mean confrontation, as you put it.