Parallel Chords in Crossing Circles

Here's a problem from an old geometry book by H. S. Hall and F. H. Stevens. The book is an augmented edition of Euclid's Elements, I - VI, with parts from book XI. For every Euclid's proposition, the authors he authors added a list of problems that illustrate or make use of that proposition. The problem at hand relates to Euclid III.3 (problem 12, p. 171):

If two circles cut one another, any two parallel straight lines drawn through the points of intersection to cut the circles, are equal.

(In the applet, the two circles are defined by three points each: P, Q, A, for one, and P, Q, D, for the other. Dragging either P or Q modifies the circles. Dragging either A or D may have different effect depending on which of the buttons at the bottom of the applet is checked. If it's "Adjust circles" then the circles will be modified. If the "Adjust chord" button is checked, A and B would be dragged over existing circles supplying a set of possible locations for the segments AB and CD.

This applet requires Sun's Java VM 2 which your browser may perceive as a popup. Which it is not. If you want to see the applet work, visit Sun's website at https://www.java.com/en/download/index.jsp, download and install Java VM and enjoy the applet.

What if applet does not run?

Solution

References

H. S. Hall, F. H. Stevens, A text-book of Euclid's elements for the use of schools, MacMillan & Co, 1904

If two circles cut one another, any two parallel straight lines drawn through the points of intersection to cut the circle, are equal.

There is of course a reason why the problem was devised to illustrate Euclid III.3:

If a straight line passing through the center of a circle bisects a straight line not passing through the center, then it also cuts it at right angles; and if it cuts it at right angles, then it also bisects it.

Drop perpendiculars from the centers E and F of the circles on the lines AB and CD (check the applet for the notations.) Since the two lines are parallel, the feet of the perpendicualrs form a rectangle UVXW. Points U, V, W, X are the midpoints of the chords AP, CQ, BP, DQ, respectively. UW = UP + PW and VX = VQ + QX. The two relations always hold if the segments involved are consdered signed. Otherwsie, three cases must be separated, say, UW = UP + PW, UW = UP - PW, and UW = PW - UP. In the first case, AB = 2 UW = 2 VX = CD. The other two cases are left as an exercise.

This applet requires Sun's Java VM 2 which your browser may perceive as a popup. Which it is not. If you want to see the applet work, visit Sun's website at https://www.java.com/en/download/index.jsp, download and install Java VM and enjoy the applet.

What if applet does not run?

This problem is very helpful in solving a related exercise.

Acknowledgement

The remarkable book by H. S. Hall and F. H. Stevens was brought to my attention by Hubert Shutrick who quoted from his private correspondence with Gordon Walsh. The book is quite remarkable in its treatment of Euclid's Elements in part because it is was published at the time when most educators began to search for ways to exclude the Elements from school. H. S. Hall and F. H. Stevens interspersed Euclid's propositions with a multitude of exercises that help students internalize Euclid's logic and master his techniques. The book shows how the Elements could be incorporated into a school program. I would prefer it to many a textbook written in the century since.

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