Can the Right Bisectors of a Triangle Meet at a Common Point?

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Homework Help Overview

The discussion revolves around proving that the right bisectors of the sides of triangle ABC meet at a common point, known as the circumcentre. The triangle is defined by its vertices A(0,a), B(0,0), and C(b,c).

Discussion Character

  • Exploratory, Conceptual clarification, Mathematical reasoning

Approaches and Questions Raised

  • Participants discuss using midpoints to establish relationships between the bisectors. There are inquiries about whether to introduce new points or use the circumcentre directly. Questions are raised regarding finding the equations of the perpendicular bisectors and the slopes of the triangle's sides.

Discussion Status

The discussion is active, with participants exploring different methods to approach the problem. Some guidance has been offered regarding the use of slopes and midpoints, but no consensus has been reached on a specific method or solution.

Contextual Notes

There is a focus on the geometric properties of the triangle and the relationships between its sides and bisectors. Participants are considering the implications of using midpoints and slopes in their reasoning.

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Homework Statement


In triangle ABC, with vertices A(0,a), B(0,0) and C(b,c) prove that the right bisectors of the sides meet at a common point (the circumcentre).


Homework Equations


Midpoint(x1 + x2 / 2 , y1 + y2 / 2)
Length of a Line

The Attempt at a Solution


I was thinking of using the Midpoints to prove that Midpoint AD = Midpoint BE = Midpoint CF...is this the right way?
 
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Try finding the equations of the perpendicular bisectors.
 
am i supposed to use new points D, E, and F? or should i use the circumcentre P?
 
You can find the slopes of the lines that make up the 3 sides of the triangle, right? Once you do that, do you know how to find the slopes of lines perpendicular to each of these three lines?

You also have one point on each of the bisectors: the midpoints of the sides of the triangles. Do you know a way of finding the equation of a line knowing its slope and one point on it?
 

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