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Home » Triangles » Triangle Inequalities » Triangle Inequality Theorem

Triangle Inequality Theorem

Last updated: Mar 27, 2021 by Ido Sarig · This website generates income via ads and uses cookies · Terms of use · Privacy policy

We all know that "the shortest distance between two points is a straight line." That is the informal way of describing the Triangle Inequality Theorem.

If we want to go from point A to point B, getting there through any intermediate point C, which is not on the straight line between A and B, is going to be longer. This should seem intuitively correct. But we will prove it formally.

Where do triangles come into this? Imagine that the line AB is a side of a triangle, ΔABC. Going from A to B is the straight, short way. Going there by first going to another point, C (which is the side AC) and then from C to B (which is the side CB) is going to be longer.

Triangle inequality theorem - A to B is shortest

Problem

In a triangle ΔABC, show that AB+AC> BC, AB+BC>AC and AC+CB>AB

Strategy

So how do we prove the Triangle Inequality Theorem?

We have already proven that the shortest distance between a point P, and a line, l, is the perpendicular line from P to l.

This problem sounds similar, so we will try to use that shortest distance theorem. To do that, we need a perpendicular line, so let's draw it:

Triangle Inequality Theorem: draw the height

This looks really similar to:

shortest distance theorem

So we can immediately see that AB>BD, and also that AC>CD.

So if we use the shortest distance theorem twice, for the two triangles created by drawing the height, we will have our proof.

Proof

(1) AD ⊥ BC     //given, we constructed AD as the height from A to BC

(2) AB > BD      //shortest distance theorem

(3) AC > CD      //shortest distance theorem

(4) AB+AC > BD+CD   //Add (2) and (3)

(5) BD+CD = BC          //given

(6) AB+AC>BC            //substitute (5) into (4)

And we can repeat this process, drawing the height from B to AC, and from C to AB to prove the other two statements.

« Shortest Distance Theorem
Deltoids: One of the Diagonals Bisects the Angles at its Endpoints »

About the Author

Ido Sarig is a high-tech executive with a BSc degree in Computer Engineering. His goal is to help you develop a better way to approach and solve geometry problems. You can contact him at [email protected]

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Welcome to Geometry Help! I'm Ido Sarig, a high-tech executive with a BSc degree in Computer Engineering and an MBA degree in Management of Technology. I'm here to tell you that geometry doesn't have to be so hard! My goal with this website is to help you develop a better way to approach and solve geometry problems, even if spatial awareness is not your strongest quality. Read More…

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