A Problem on Locus Here is a locus problem you can either attempt to prove theoretically,...

A Problem on Locus Here is a locus problem you can either attempt to prove theoretically, or try it out on Sketchpad. There is actually an item in the CON-STRUCT menu called Locus, and it can be used to solve many locus problems. Consider this: Start with any isosceles triangle, ∆ABC, and points X and Y on its legs and such that AX = AY. Find the locus of P, the point of intersection of and as X varies on . An outline of the steps needed in Sketchpad are as follows:

[1] (Turn Autoshow Label off.) Construct any segment , locate its midpoint, and construct the perpendicular at that midpoint. Select a point A on the perpendicular, and construct segments and . Hide the perpendicular and the midpoint. The vertex of the isosceles triangle created can be dragged, as well as points B and C, to create any shape for the isosceles triangle.

[2] Select any point X on , and draw segment .

[3] Construct the Circle By Center and Point having center A and passing through X. Select the point of intersection of this circle with . This will be point Y in the locus problem mentioned.

(4] Obtain the point P of intersection of segments and . To activate the Locus command properly, select point X, then point P (in that order), and choose Locus under CONSTRUCT.

Did anything predictable happen? Can the result be proven geometrically? What do you think the locus is? Is this feature invariant when you drag points A and B? For further investigation, try this with a nonisosceles triangle.