Problem
GEO-B2-M09-P009 Circle of a Right Triangle
#9
★★☆☆☆ Level 2 of 5
Find the equation of the circle through \(A(0,0)\), \(B(a,0)\), \(C(0,b)\), and prove that its centre is the midpoint of \(BC\).
Use the form \(x^2+y^2+ux+vy=0\).
Substituting \(B(a,0)\) gives \(u=-a\), and substituting \(C(0,b)\) gives \(v=-b\). The circle is \(x^2+y^2-ax-by=0\). Completing squares gives \(\left(x-\frac a2\right)^2+\left(y-\frac b2\right)^2=\frac{a^2+b^2}{4}\). The centre is \(\left(\frac a2,\frac b2\right)\), the midpoint of \(BC\).
Connects the equation of a circle with a geometric fact.