三角不等式的逆命题：满足三角不等式的三边是否可构成三角形？

Absolutely! If you have three positive real numbers that satisfy the triangle inequality, you can always construct a triangle with those numbers as its side lengths. Let me break down the proof clearly for you.

First, let's formalize our starting conditions:
Let (a, b, c) be positive real numbers such that all three triangle inequalities hold:

• (a + b > c)
• (a + c > b)
• (b + c > a)

Step 1: Set up a coordinate system #

We'll use a simple geometric construction to build the triangle:

• Place point (A) at the origin ((0, 0)) of a 2D coordinate plane.
• Place point (B) at ((c, 0)) — this makes the length of (AB) equal to (c).

Step 2: Define the loci for point (C) #

We need a point (C) such that:

• The distance from (A) to (C) is (b) (so (AC = b))
• The distance from (B) to (C) is (a) (so (BC = a))

These conditions mean:

• (C) lies on the circle centered at (A) with radius (b)
• (C) lies on the circle centered at (B) with radius (a)

Step 3: Prove the two circles intersect #

For two circles to have at least one intersection point, two key conditions must be met:

1. The sum of their radii is greater than the distance between their centers: (a + b > c)
2. The absolute difference of their radii is less than the distance between their centers: (|a - b| < c)

Both of these are exactly our given triangle inequality conditions!

• The first condition (a + b > c) is provided directly.
• The second condition (|a - b| < c) expands to two inequalities we already know:
  • (a - b < c) → (a + c > b)
  • (b - a < c) → (b + c > a)

Since both intersection requirements are satisfied, the two circles must cross at at least one point (C).

Step 4: Confirm the triangle exists #

Once we have point (C), the triangle (ABC) has sides:

• (AB = c)
• (AC = b)
• (BC = a)

Which is exactly the triangle we wanted to construct.

A quick note on positivity #

We specified (a, b, c) are positive because side lengths can't be zero or negative—if any value were zero, you'd end up with a line segment instead of a triangle, and negative lengths have no geometric meaning.

内容的提问来源于stack exchange，提问作者Юрій Ярош