Wikipedia:Reference desk/Archives/Mathematics/2015 January 21

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January 21

Triangles and circumcenters

My points are (-3,1) (-1,-1) (4,-2)

I graphed the triangle and found all the perpendicular bisectors. I got them all to connect at a point and that's the circumcenter.

How do you solve for this algebraically? — Preceding unsigned comment added by 124.180.249.108 (talk) 00:20, 21 January 2015 (UTC)

Homework? There's probably a nicer way to do this, but a brute force approach should work... Select two points, work out the gradient of the line between them, then find their midpoint, and write down the equation for the perpendicular running through that midpoint (recall that the product of the gradients of two perpendicular lines is -1). Repeat for another pair of points. Now you have a pair of simultaneous linear equations. Solve for their intersection. RomanSpa (talk) 01:22, 21 January 2015 (UTC)

P.S. I hope there's a more elegant approach, and that someone here will remind me what it is! Thanks. RomanSpa (talk) 01:22, 21 January 2015 (UTC)

That's how I'd do it, too. StuRat (talk) 06:06, 21 January 2015 (UTC)

I would plug vertices' $(x,y)$ coordinates into the general circle equation $(x-a)^{2}+(y-b)^{2}=r^{2}$, eliminate $r^{2}$, then expand squares and reduce $a^{2}$ and $b^{2}$ to obtain a 2×2 linear equation system with $a,b$ unknown. --CiaPan (talk) 06:20, 21 January 2015 (UTC)

That's much neater than my way. Interestingly, your approach requires us to know that the intersection of the perpendiculars is the circumcenter (presumably from the geometric proof), while mine/StuRat's doesn't require this knowledge. Once we have the intersection of the perpendiculars it's easy to prove algebraically that this is the circumcenter, of course (hint for the original questioner: it's just Pythagoras...). I suspect this was someone's homework question, and now we can see why: we can prove that the circumcenter is the intersection of the perpendiculars geometrically, and I suspect our original questioner has already seen this proof; now we see that we can prove the same thing algebraically. That is, one area of mathematics can be mapped onto another area - here we're mapping geometry to algebra - and proving something in one area can (with a suitable choice of mapping) prove something in another area. This is an important technique in more advanced mathematics, and I suspect our original questioner is just encountering this idea for the first time. RomanSpa (talk) 13:19, 21 January 2015 (UTC)

Given a triangle with vertices (x1, y1), (x2, y2), (x3, y3), the equation of the circumcircle is given by the determinant

${\begin{vmatrix}x^{2}+y^{2}&x&y&1\\x1^{2}+y1^{2}&x1&y1&1\\x2^{2}+y2^{2}&x2&y2&1\\x3^{2}+y3^{2}&x3&y3&1\end{vmatrix}}=0.$

You can then find the coordinates of the center by completing squares. The final expression is a bit messy when you expand everything out, a fraction with 12 term numerator and 6 term denominator for each coordinate. --RDBury (talk) 10:36, 21 January 2015 (UTC)