[b]Algebraic Proof of the Witch of Agnesi[/b][br][br]The Witch of Agnesi is a curve defined using a circle with diameter OM, where O is the origin (0,0) and M is the point (0,2a) on the positive y-axis. The construction leads to the following Cartesian equation of the[br]curve: y = 8a[sup]3[/sup] / (x[sup]2[/sup] +4a[sup]2[/sup])[br][br]When a = 1/2, the equation simplifies as follows: y = 8(1/2)[sup]3[/sup] /(x[sup]2[/sup] + 4(1/2)[sup]2[/sup]) = 1 / (x[sup]2[/sup] + 1)[br][br]Equivalently, by multiplying through by the denominator, we obtain: (x[sup]2[/sup] + 1) y = 1[br][br]This simplified form shows that the Witch of Agnesi is the graph of the derivative of the arctangent function, since d/dx(arctan(x)) = 1/ (1 + x[sup]2[/sup]).[br][br][b]Parametric Representation[/b][br]The Witch of Agnesi can also be represented parametrically,[br]where θ is the angle between OM and OA, measured clockwise:[br]    x = 2a tan(θ)[br]    y = 2a cos²(θ)[br][br]Thus, the Witch of Agnesi can be studied both algebraically[br]and geometrically, revealing deep connections between circle geometry, rational[br]functions, and calculus.[br][br][b]Drag Test:[/b][br]Drag [b]θ[/b] and watch point [b]P[/b] trace out the Witch.[br]Adjust [b]a[/b] to see the curve scale.