**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.