The triangle inequality is one of the simplest geometric inequalities. It conveys a very simple idea, that in (Euclidean) space, the straight line distance is the shortest path between two points. Specifically, if and are side lengths of a triangle, then . The inequality is strict if the triangle is non-degenerate (has non-zero area).
There are several proofs of the triangle inequality; one proof uses AM-GM, while another uses cosine rule and the fact that .
There are several useful consequences of the triangle inequality:
A. In a (non-degenerate) convex quadrilateral ,
Proof: Let intersect at . From the triangle inequality on and , and . Thus, Equality holds if and only if and are straight lines, which forces to be a straight line.
B. If is a point in the interior of triangle , then
Proof: Extend to intersect at . From the triangle inequality on and , we get and . Then, Equality holds if and only if and are straight lines, which forces .
1. If the lengths of a non-degenerate triangle are 9 and 15, what are the possible lengths of the third side?.
Let the third side have length . The 3 sides must satisfy and . The third inequality is trivially true, and the first two give .
2. If is a point in the interior of triangle , show that
Apply the triangle inequality to triangles and to obtain and . Adding up these 3 inequalities and dividing by 2 gives the inequality on the left.
Apply consequence B thrice to obtain and . Adding these 3 inequalities and dividing by 2 gives the inequality on the right.
3. In triangle , is the midpoint of side . Show that
Extend to such that . Since , by Parallel lines Property D, triangles and are similar with ratio 2, which gives . Applying the triangle inequality to , we get . Substituting the previous equations give . Equality holds if points and are on the same line, which implies that points and are on the same line, which is a degenerate triangle.
4. All the internal diagonals of a convex polygon have equal lengths. What is the maximum number of sides that the polygon can have?
The regular pentagon clearly satisfies the condition. We will show by contradiction that any convex polygon with sides will not satisfy the condition. Suppose not, let be such a polygon. Label the vertices in (clockwise) order. Then form a convex quadrilateral, so by consequence A, . But the condition given states that all the internal diagonals have equal length, so , which is a contradiction. Hence, , and we know that 5 sides is attainable.