Geometric Proof: Triangle Inequality Theorem for Point O | Homework Help"

[Michael_Light]

Homework Statement

If O is any point inside a triangle ABC, prove that BA + AC > BO + OC.

Homework Equations

The Attempt at a Solution

Any hints? Thanks...

 

[lewando]

Since you are looking to prove an inequality, you should consider exploring the (not sure what you call it) Sum of Two Sides of a Triangle is Greater than the Third Side theorem. It also might help to draw a line segment from B through O to a point D on AC.

 

[ArcanaNoir]

Since you are looking to prove an inequality, you should consider exploring the (not sure what you call it) Sum of Two Sides of a Triangle is Greater than the Third Side theorem.

AKA "the triangle inequality"

 

[Michael_Light]

Still cannot do it... I tried to apply the triangle inequality theorem and extending those lines, but yet, still cannot prove it.

I am so clueless, I need more hints..

 

[lewando]

I thought the hint about extending that line segment would be pretty big. What it does is give you a "bridge" between triangle ABC and BOC. The brigde triangle, BDC, can be related to the other triangles, I guess by using the so-called triangle inequality .

 

[lewando]

So try this approach: Do it in steps: first step: only consider BOC and BDC. Write down all the facts you know about this configuration.

Can you prove this intermediate statement: BD + DC > BO + OC? Try to do this by using the "algebra" of geometry (adding/subtracting the same thing to both sides does not change the truth of the equation/inequality), using your facts to get to the intermediate statement you are trying to prove. Or take the intermediate statement and decompose it into one or more combinations of your facts.

Good luck. If this wasn't hard, it'd be easy.