**THEOREM E**:
Let *J* be any simple closed curve, *T* any triangle, and
let *V* be the set of points on *J* that are vertex to a triangle
inscribed in *J* that is similar to *T*. Then the set
*V* is dense in *J* (meaning that
there are points of *V* arbitrarily
close to any given point of *J*).

*Proof:*
Let *A, B,* and *C* be the vertices of *T*.
Let *P* be a point on *J*. Define a function

*f(P) = P*,- If
*Q*is a point other than*P*then*f(Q)*is the point so that triangle*PQf(Q)*is directly similar to*T*with*P*corresponding to*A*,*Q*corresponding to*B*, and*f(Q)*corresponding to*C*.

The image *f(J)* of the curve *J* will itself be a curve
passing through the point *P*. If *J* is reasonably well-behaved
at *P* these curves will cross each other there (see the first diagram).

But if *f(J)* crosses *J* at *P*, then part of *f(J)*
lies *inside* of *J* while part lies *outside* of *J*.
So, *f(J)* must meet *J* at another point somewhere.
This means there is a point *f(R)* on *f(J)* that is also on
*J*. But then the points *P, R, * and *f(R)* all lie on
*J* and form a triangle similar to *T*. (See the second diagram.)

So, any point *P* that is nice enough to make *J* and
*f(J)* cross will be a vertex to an inscribed triangle similar to
*T*.
It takes some topological arguing (the details of which we won't give here),
but it can be proved that the candidates for such a well-behaved *P*
are common enough to be dense in the curve *J* -- that is, there is
always such a well-behaved point as close as you want to any point on
*J*.

Note: Mark Meyerson has proved a remarkable
fact that strengthens this theorem in the case that *T* is an
equilateral triangle:

[Mark D. Meyerson,

It seems likely that something like Meyerson's Theorem is actually true for all triangles. This would strenthen Theorem E considerably!