Triangle with inscribed circle

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Homework Help Overview

The problem involves a triangle ABC with an inscribed circle of radius greater than 1, and the goal is to prove that at least one of the distances from a point P inside the triangle to the vertices (PA, PB, or PC) is greater than 2. The subject area is geometry, specifically dealing with properties of triangles and circles.

Discussion Character

  • Exploratory, Assumption checking, Conceptual clarification

Approaches and Questions Raised

  • Participants discuss the implications of the distances PA, PB, and PC being less than or equal to 2, and how that relates to the circumradius. There are suggestions to consider area formulas for the triangle and the inscribed circle, as well as the potential utility of differentiating a minimum function.

Discussion Status

The discussion is ongoing, with participants exploring different approaches and questioning the assumptions made about the distances and areas involved. Some guidance has been offered regarding potential formulas and relationships, but no consensus has been reached on a specific method to prove the statement.

Contextual Notes

The problem is derived from a Russian geometry textbook, and there is a noted lack of clarity regarding the class context, which may affect the direction of the discussion.

klawesyn28
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P is a point inside triangle ABC. In the triangle there is inscribed
circle which radius is greater than 1. Prove that PA>2, PB>2 or PC>2.


I don't know how to solve it. Could anybody help me?
 
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Did you at least start the problem? If so then post what you have and we will help you from there.
 
Yes I started with this problem and then I don't know how to prove that when PA<=2, PB<=2, PC<=2, the circumradius is smaller than 2. P is point inside the triangle.
 
klawesyn28 said:
P is a point inside triangle ABC. In the triangle there is inscribed
circle which radius is greater than 1. Prove that PA>2, PB>2 or PC>2.
I don't know how to solve it. Could anybody help me?

This is one of those cases where the first step can be tough. You haven't said what kind of class this is for and so it's hard for us to figure out what kind of direction you're supposed to go in.

With that, here are some things you might think about.

(a) It would be really handy if you had a formula for the minimum that you could differentiate. It might be useful for you to know that:

[tex]min(A,B) = (|A+B| - |A-B|)/2[/tex]

(b) Do you think you could do something with the area formula for a triangle as compared to the area formula for the circle? Can you write an area formula that is defined around the point P? Perhaps the areas of the three triangles PAB, PBC, PCA?

Carl
 
It's task from russian books with task from geometry. I don't see how your hints can help me.
 

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