Lemma:

There are two bars  RN and PA and are joined at the point R. K is the midpoint of P and R., then the locus of K is an ellipse as A travel through a straight line.

Pf :

Let the coordinates of N、A、P and K are (0, 0)、(x, 0)、(0, y) and (s, t) each other and the length of  RN is a . On the other hand, the coordinate of K is . Hence and . Since the distance of  is 2a, we get  or .

In fact, it is an ellipse.

Theorem:

In the linkage given here with the conditions obtain: M and N are fixed points to which the whole system is pivoted.

Pf:
From construction 11 of straight line, the point A moves through a straight line.
In combination with the above, the locus of K is an ellipse as B move.

Use for linkage:

If we fix the points M and N on the plane，then K will describe an ellipse.

A  New Linkage for Describing a straight line by continuous Motion ,
John James Quinn,
American Mathematicl Monthly, Volume 16, Issue1(Jan.,1909).