Is there an easier way to solve a "Find the locus" problem?
Note: I am not concerned with the accuracy of my solution so you don't need to redo any of my calculations.
Original question: Suppose ABC is an equilateral triangular lamina of side length unity, resting in two-dimensions. If A and B were constrained to move on the x- and y-axis respectively, then what is the locus of the centre of the triangle?
My solution: Let the vertices of be , and . Let the centre of the triangle be . We wish to find the locus of D under the constraints:
Since D is the centre of the triangle, we know
Parametrizing: Let be the angle that the edge AB makes with the x-axis. We use (theta) as our parameter of choice to write,
We can parametrize this with another parameter as
Plugging these back into third equation gives
After lots of algebraic manipulations...