### Doris Schattschneider, Moravian College

Parametric equations of curves is a main topic in calculus (usually the second semester). Many of the curves for which parametric equations are sought are generated as loci of moving points that are constructed geometrically. The cycloid is a well-known example treated in most calculus texts. Our text, Stewart's Calculus, treats this in section 9.1. At the end of that section are several problems in which a sketch of a geometrically constructed point is given and the student is asked to find the parametric equations of the locus of the point, and also identify the locus.

#### Problem 35

The sketch for problem 35 is given. The question "What is the locus of P, as the ray OQ swings through a full revolution?" is easily answered by dragging Q around the circle in the dynamic sketch (or you can animate Q on the large circle). You can also change the size of each of the circles by dragging Q or S on the radii that are shown in the upper left of the sketch.The student must still find the parametric equations for the locus of P as the ray OQ sweeps around the circle, where the parameter is the angle with initial side the positive x-axis, and terminal side OQ.

#### Problem 40

The sketch for Section 9.1, problem 40 is given. The curve generated by P when point B is animated on the line x = 2a is called the "cissoid of Diocles". It is helpful to generate the curve with Sketchpad as well as discover the constraints on how far B must travel to sweep out the whole curve. The student must still find the parametric equations for the locus of P as B travels back and forth on the line.

### A locus problem from a computational geometer

Given fixed points A, B and a line, construct circles through A and B that are tangent to the line, if possible. What is the locus of the points of tangency as the line moves up or down (with A and B remaining fixed)?

This geometry problem arose when Scott Drysdale, a computer scientist, wanted to implement an algorithm to construct a Delauney triangulation of a set of points in the plane, using the "empty circle" technique. Once the construction has been made with Sketchpad, it is easy to see the answer to the question above. It took Drysdale and a colleague more than a week to find the answer analytically.

Drag the line up and down to trace the locus of the points of tangency of the circles.

### Grunbaum's Exploration

The perpendicular bisectors of the sides of quadrilateral ABCD form a quadrilateral Q (unless ABCD is a rectangle), and the perpendicular bisectors of the sides of Q form a quadrilateral Q'. Show that Q' is similar to Q.

This problem was proposed by Josef Langr in the American Mathematical Monthly in 1953 [Problem E 1050, volume 60, p. 551], and brought to our attention by Branko Grunbaum in an article "Quadrangles, Pentagons, and Computers," in Geombinatorics 3 (1993) 4-9. No solution to the problem has ever been published.

The sketch shows ABCD with Q in outline and Q' shaded. Lines have been constructed through what appear to be corresponding vertices of ABCD and Q'. The lines are concurrent at point P, and with Sketchpad, you can perform a dilation with center P and ratio ±(edge of ABCD/corresponding edge of Q') that sends Q' onto ABCD. (To mark the ratio, first choose an edge of Q', then the corresponding edge of ABCD.) The ratio is negative when ABCD is convex, and positive when ABCD is nonconvex. Remember that a dilation with a negative ratio is the composite of a halfturn through P followed by the dilation through P with the corresponding positive ratio. By dragging any vertex of ABCD, the similarity of Q' to ABCD is maintained - even in the self-intersecting cases. So Sketchpad "proves" the theorem - we still await a conventional proof.

Grunbaum challenges us to find a full proof of Langr's problem and determine the ratio of the dilation which somehow depends on the shape of ABCD. Other problems to be solved: (a) determine for what quadrilaterals ABCD the quadrilateral Q is similar to ABCD; (b) investigate what happens with pentagons and with hexagons.

Do we call Langr's problem a theorem? Is the strong evidence provided by Sketchpad enough to declare that there is a theorem, even though we don't have a conventional proof?

### Notes by Jim King on Grunbaum's exploration

Branko Grunbaum has recently written about a problem of Langr which is easy to explore with Sketchpad.

Begin with a quadrilateral Q. Define a new quadrilateral Q1 whose sides are the perpendicular bisectors of the sides of Q, and then define Q2 to be the quadrilateral whose sides are the perpendicular bisectors of Q1.

It is easy to do this with Sketchpad and to observe visually that Q2 is always a dilation of Q. However, the proof of the theorem is another matter. No published elementary proof is known, although this result is listed by Chou in his book as one of the theorems proved by an automatic theorem-proving program (S-C Chou, Mechanical Geometry Theorem Proving). No published proof of the ratio of dilation is known.

In addition, Grunbaum has observed in computer examples that the analogous construction for pentagons, beginning with P and then constructing P1, P2, P3, ... by taking perpendicular bisectors of the sides, always constructs P3 a dilation of P1 (though not P2 a dilation of P). Schattschneider points out that this raises an interesting question: if a geometric result of this nature is established in enough examples to convince mathematicians that it must be true, then is it a theorem or must one await a traditional proof?

(Note: Motivated by Schattschneider's talk, Dan Bennett and Jim King independently found an elementary proof that Q2 is a dilation or translation of Q, but with no information about the ratio. G.C. Shephard, in a preprint called "The Perpendicular Bisector Construction," proves by trigonometry that the second perpendicular quad Q2 is homothetic (a dilation) of Q and also gives a formula in terms to trig functions for the ratio of similitude. The formula is quite complicated. There is no obvious geometric interpretation of the ratio; the formula is not even obviously symmetric in the order of vertex angles.)