The Triangle Inequality Theorem is fundamental in geometry. For any three lengths to form a triangle, they must satisfy a specific condition: the sum of any two sides must be greater than the third side. This gives us three inequalities that must all be true.
The Triangle Inequality gives us three specific conditions. First, a plus b must be greater than c. Second, a plus c must be greater than b. Third, b plus c must be greater than a. Each inequality ensures that no single side is too long compared to the other two.
Let's test the famous 3-4-5 triangle. First, 3 plus 4 equals 7, which is greater than 5. Second, 3 plus 5 equals 8, which is greater than 4. Third, 4 plus 5 equals 9, which is greater than 3. Since all three conditions are satisfied, these sides can form a valid triangle.
Now let's try sides of length 1, 2, and 5. First, 1 plus 2 equals 3, which is less than 5. This violates the triangle inequality! Even though the other two conditions pass - 1 plus 5 is greater than 2, and 2 plus 5 is greater than 1 - just one failure is enough. These three lengths cannot form a triangle.
三角形不等式定理为我们提供了一个简单的算法来检验任意三个长度能否构成三角形。首先，检查a加b是否大于c，如果不是，则不能构成三角形。如果是，继续检查a加c是否大于b，如果不是，则不能构成。最后，检查b加c是否大于a。只有当这三个条件都满足时，三条边才能构成一个有效的三角形。这个检验方法在几何学中是基础的，在计算机图形学、工程学和许多其他领域都有应用。