Rotating Triangles

By

Jeffrey R. Frye

Problem:  What is the locus of the third vertex of a triangle when its first two vertices are moved along the x and y axes respectively?

The first challenge is to find a way to complete the construction that will allow rotating triangles to be displayed.  There are several ways that this can be accomplished.  One way is to place a free point on either the x-axis or on the y-axis.  Once this point has been placed, it will be the location of one of the triangle vertices.  From this point, construct a segment of a specified length that has its endpoint on the other axis.  If the free point is on the x-axis, then this point will be on the y-axis.  By using the properties of triangles, we can construct segments of specific lengths that will locate the third vertex that it is not on either axis.  The picture below shows this construction.

By tracing the third vertex, it appears that a conic is traced.  What is the locus of this vertex?  Click here to explore with GSP.

As an additional exploration, a free point can be placed on the side of the triangle that connects the vertices that are on the axes.  This point will also trace the same type of conic.  It will be oriented to the coordinate axis system.  Click here for a different GSP that shows this.  By drawing similar triangles and establishing the proportional relationships, the equation that describes an ellipse can be derived.  The construction of the similar triangles is shown below.  The equation explanation follows.