The circumcenter is defined as the point at which the three perpendicular bisectors of a triangle meet. Let's prove that the three perpendicular bisectors of any triangle really do meet in a single point.
First, let's find equations for the perpendicular bisectors of a triangle.
By our calculations for the orthocenter, we know
is vertical,
the slope of 
is 
,
and the slope of 
is 
.
passes through the point (a, 0), so is the line described by
x = a.
passes through the point (b, c), so is the line described by
passes through the point (a + b, c), so is the line described by
Let's find the point 
at which and intersect, and then we can determine whether lies on as well. If it does, we will have shown that all three perpendicular bisectors intersect in a single point.
We need to find a point that is on both and
The y-coordinate of both lines must be the same at x = 
.
So we must have:
And
Does our point lie on ? Let's find out.
The x-coordinate of any point on is a.
= a, so does indeed lie on .
|
So for any triangle, all three perpendicular bisectors intersect at a single point, the circumcenter.
For a triangle whose vertices are (0,0), (2a,0), and (2b,2c), the coordinates of the circumcenter are
|
<< Previous (Orthocenter)
Return to my page
Return to EMAT 6680 page
