唇亡齿寒 lips gone, teeth cold
Explanation
这个成语比喻关系密切，利害相关，一方有难，另一方也会受到损害。
This idiom refers to a close relationship, where both sides are dependent on each other's fate. If one is in trouble, the other will suffer as well.
Origin Story
春秋时期，晋国为了攻打虢国，晋王采纳大将荀息的建议，送晋王的玉石与宝马给虞王，向虞国借道攻虢。虞王不接受宫之奇邻国之间就是嘴唇与牙齿的关系的观点，允许晋国借道攻虢，结果晋灭虢后三年就消灭虞国，夺回宝物。这个故事告诉我们，国家之间，即使没有直接的利益关系，也要互相帮助，否则，一旦一方受到伤害，另一方也会跟着遭殃。
During the Spring and Autumn Period, the state of Jin wanted to attack the state of Guo. The King of Jin followed the advice of his general Xun Xi and sent the King of Yu jade and horses as gifts to secure passage through the state of Yu to attack Guo. The King of Yu rejected the advice of Gong Zhi Qi, who said that the relationship between neighboring states was like that of lips and teeth. He allowed Jin to pass through Yu to attack Guo. Three years later, Jin destroyed Guo and recovered the gifts. The story teaches us that even if there are no direct interests between countries, they should help each other. Otherwise, if one is harmed, the other will suffer as well.
Usage
这个成语通常用来比喻两个事物之间关系密切，一荣俱荣，一损俱损。
This idiom is often used to illustrate that two things are closely linked and affect each other. If one is doing well, the other is also doing well. If one is doing badly, the other is also doing badly.
Examples
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两国之间唇亡齿寒，休戚相关。
liang guo zhi jian chun wang chi han, xiu qi xiang guan
Two countries are closely linked and share the same fate.
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企业之间相互依存，唇亡齿寒，要共同维护彼此利益
qi ye zhi jian xiang hu yi zhun, chun wang chi han, yao gong tong wei hu bi ci li yi
Companies are interdependent and need to support each other because they are like gears that mesh together and cannot function without each other