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

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In triangle ABC with vertices A(0,a), B(0,0), and C(b,c), the discussion focuses on proving that the right bisectors of the triangle's sides intersect at a common point, known as the circumcentre. The approach involves calculating the midpoints of the sides and deriving the equations of the perpendicular bisectors. Participants suggest using the slopes of the triangle's sides to find the slopes of the perpendicular lines. The discussion emphasizes the importance of knowing a point on each bisector and how to formulate the line equations using this information. Ultimately, the goal is to demonstrate that these bisectors converge at a single point.
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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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