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Triangle Circumcenter's Coordinates

Date: 08/15/2012 at 19:26:01
From: Dawn
Subject: coordinate proof for circumcenter of a triangle

Triangle ROS is drawn on the coordinate plane. Vertex O is at the origin
(0, 0) and side OS lies on the x-axis. The coordinates of point S are 
(6a, 0) and the coordinates of point R are (6b, 6c).

Prove that the perpendicular bisectors of the three sides of triangle ROS
meet at circumcenter C with coordinates

   (3a, (3b^2 + 3c^2 - 3ab)/c)

I can't quite get to the final solution of this.

I did find the midpoints of each side of the triangle:

   (3a, 0), (3b, 3c) and (3a + 3b, 3c)

I've also found the slope of RS:

   c/(b - a)

So the slope of its perpendicular bisector is (a - b)/c.

I also found the slope of OR:

   c/b

So the slope of its perpendicular bisector is -b/c.

I got the equation of the perpendicular bisector of OR as

   y = (3c^2 - bx + 3b^2)/c.

I tried writing the equation for the perpendicular bisectors of RS and OR,
using point-slope form and knowing the midpoint, which is on the line and
the slope of the line. I had a feeling that simultaneously solving these
would give the answer; but when I tried, it just got hopelessly
complicated!

And that's as far as I got.

I do know what circumcenter means -- that that point is equidistant from
each vertex.



Date: 08/16/2012 at 13:00:33
From: Doctor Schwa
Subject: Re: coordinate proof for circumcenter of a triangle

Hi Dawn,

That's some great work!

You're almost done, and you have all the right ideas. You need to find the
intersection of two perpendicular bisectors to figure out where the
circumcenter is -- just one more equation.

It looks to me like the perpendicular bisector of OS has a simpler
equation. Don't try to use point-slope for it! Once you have that
equation, it'll be pretty quick to substitute its result into your
equation for OR and find the coordinates of the intersection point.

Give that a try, and please let us know if you have further questions!

- Doctor Schwa, The Math Forum
  http://mathforum.org/dr.math/ 



Date: 08/16/2012 at 22:50:04
From: Dawn
Subject: Thank you (coordinate proof for circumcenter of a triangle)

Dr. Schwa:

Thank you so much!

It's amazing how you can stare at a problem for so long you just don't 
see the simplest things. Duh! That's what I get for trying to finish it 
at 3 am.

Thanks for putting me on the right track. I got it! 

You guys are awesome!

Dawn



Date: 08/16/2012 at 23:50:24
From: Doctor Schwa
Subject: Re: Thank you (coordinate proof for circumcenter of a triangle)

Hi Dawn,

No problem! My pleasure!

It really is great when people take the time to work on the problem and
write up their thinking so clearly. It's a lot easier for us to give
helpful advice when you do that.

Thanks,

- Doctor Schwa, The Math Forum
  http://mathforum.org/dr.math/ 
Associated Topics:
High School Coordinate Plane Geometry

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