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### Comparing Sidelengths of Nested Triangles that Share the Same Base

```Date: 09/07/2011 at 01:00:32
From: SANDEEP
Subject: geometry

Prove that, for any point O inside a triangle ABC,

AB + AC > OB + OC

I don't understand how to proceed. I know that the greater the sidelength,
the greater the angle opposite it.

```

```
Date: 09/07/2011 at 09:00:14
From: Doctor Floor
Subject: Re: geometry

Hi, Sandeep,

We will make use of the triangle inequality. See, from the Dr. Math
archives,

http://mathforum.org/library/drmath/view/55276.html

If O is in the interior of ABC, we can construct BO to intersect AC at a
point Q.

A
/ \
/  _Q
/ _O_ \
/_-   -_\
B---------C

From the triangle inequality, we know that

OC < OQ + QC

Adding OB to both sides of the inequality,

OB + OC < OB + OQ + QC
= QB + QC

From the triangle inequality, we also know that

BQ < AB + AQ

You finish.

If you have more questions, just write back.

Best regards,
- Doctor Floor, The Math Forum
http://mathforum.org/dr.math/
```
Associated Topics:
High School Euclidean/Plane Geometry

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