Proving Segment Measure in Triangle ABC: BM+CM < AB+AC

In summary, the measure of a segment is dependent on the points that make up the segment. If one of the points is moved, the segment can no longer be measured using the same methodology.
  • #1
Cantor
8
0
I have a question involving the measure of segments

If m is a point inside a triangle ABC how could we prove that segment BM+CM < AB+AC. I am trying to use the Triangle Inequality Theorem (If A, B, C are three non-collinear points then AC < AB+BC) but I am stuck.

Should I prove by contradiction or show that BM < AB and CM <AC

Any help would be appreciated, Thanks:smile:

th_26657_triangle.jpg
 
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  • #2
My first instinct is to draw some more lines. (e.g. maybe draw in the segment AM, or extend BM and CM to intersect the opposite sides of the triangle)

This would give me more inequalities to play with, and maybe I could derive something from there.


P.S. BM doesn't have to be less than AB. A counterexample isn't hard to find -- just draw AB and BM so that BM > AM, and then try and draw in the rest of the diagram.
 
  • #3
Just make sure that whatever inequalities you end up working with, knowing they are true ensures that m is inside ABC.
 
  • #4
Try thinking of points B and C as foci of an ellipse with A on the boundary.

Carl
 
  • #5
I have tried all the possible inequalities but still nothing, the problem i am having is M could be anywhere so AB could be bigger or small them BM depending. Any other ideas by any chance.:smile:

Thanks

Cantor
 
  • #6
Try thinking of points B and C as foci of an ellipse with A on the boundary.
Based on the title of the post, I think he's at a level where he's not allowed to simply assume that one ellipse lies inside the other, and he's not in a class where it's fair game to assume the Jordan curve theorem.

And the thought of studying conic sections in a (possibly) non-Euclidean setting makes me shudder! :yuck:



Anyways, back to the original question -- I think looking at degenerate cases might help. They're often easier. What if M lies at one of the vertices? Or on one of the sides?

And, obviously, the theorem isn't true if M lies outside the triangle, so you have to use some fact that only works when M is inside.

(And, there's still the chance that inspired... or brute force... manipulation of the inequalities will provide the answer)
 
Last edited:

1. How can I prove that BM+CM is less than AB+AC in Triangle ABC?

There are several ways to prove this statement. One way is to use the Triangle Inequality Theorem, which states that the sum of the lengths of any two sides of a triangle must be greater than the length of the third side. In this case, BM+CM represents the sum of the lengths of two sides, while AB+AC represents the length of the third side. Therefore, according to the Triangle Inequality Theorem, BM+CM must be less than AB+AC.

2. What other theorems or properties can I use to prove this statement?

In addition to the Triangle Inequality Theorem, you can also use the properties of parallel lines and transversals to prove this statement. For example, if you can show that BM and CM are parallel to AC, then you can use the Parallel Lines Cut by a Transversal Theorem to prove that BM+CM is equal to AC. Similarly, if you can show that BM and CM are parallel to AB, you can use the same theorem to prove that BM+CM is equal to AB.

3. Are there any other conditions or constraints that must be met in order to prove this statement?

In order to prove this statement, you must have a valid triangle. This means that the sum of any two sides must be greater than the length of the third side. If this condition is not met, the statement cannot be proven.

4. What if I cannot prove this statement using the given information?

If you are unable to prove this statement using the given information, it is possible that the statement is false. You may need to gather more information or consider alternative approaches in order to prove or disprove the statement.

5. Can this statement be proven using any other methods besides the ones mentioned?

Yes, there are other methods that can be used to prove this statement. For example, you can use the properties of similar triangles to prove that BM+CM is less than AB+AC. Additionally, you can use algebraic methods, such as the Triangle Inequality Theorem using the distance formula, to prove this statement. Ultimately, the method you choose will depend on the given information and your personal preference.

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