Patrick Honner<p>I noticed something interesting (and perhaps obvious!) while playing around with linear equations this morning.</p><p>When trying to the find the equation of a circle through three points using a system of equations, the strategy is to plug each point into</p><p>\[x^2+y^2+Ax+By+C=0\]</p><p>and to solve the resulting 3x3 system for \( A, B, \) and \(C\). Suppose two points on the circle are \( (2,3) \) and \( (4,4) \). When you plug these in, you get the two equations</p><p>\[ 2A + 3B + C = -13\]<br>\[ 4A + 4B + C = -32\]</p><p>Combining these equations gives you</p><p>\[2A + B = -19\]</p><p>Notice that the slope of the line between the two points, \( \frac{1}{2} \) is encoded in the coefficients of this new equation: it's the coefficient of \(B\) divided by the coefficient of \(A\).</p><p>This is one way to see why three collinear points don't determine a circle. When you plug in all three points to get the system of equations, the fact that the slopes between pairs of points are all the same produces a dependence on the left-hand side of the system that isn't consistent with the right-hand side. For example, if the third point is \( (10,7) \) (which is collinear with the points \( (2,3) \) and \( (4,4) \)), the 3x3 system can be reduced to the system</p><p>\[2A + B = -19\]<br>\[6A + 3B = -117 \]</p><p>which is inconsistent.</p><p>I'm sure this can be probably be understood in a cool way by thinking about duality, or the relationship between affine and linear functions, but I found this simple approach satisfying!</p><p><a href="https://mathstodon.xyz/tags/Math" class="mention hashtag" rel="nofollow noopener" target="_blank">#<span>Math</span></a> <a href="https://mathstodon.xyz/tags/LinearAlgebra" class="mention hashtag" rel="nofollow noopener" target="_blank">#<span>LinearAlgebra</span></a> <a href="https://mathstodon.xyz/tags/Mathematics" class="mention hashtag" rel="nofollow noopener" target="_blank">#<span>Mathematics</span></a></p>