# Van Schooten's Locus Problem

According to [H. Dorrie, #47], Franciscus van Schooten (1615-1660), a Dutch mathematician, treated the following problem in his *Exercitationes mathematica*, published in 1657:

Two vertices of a rigid triangle in a plane slide along two arms of an angle of the plane; what locus does the third vertex describe?

J. Casey in his *Analytic Geometry* (1885) also ascribes the problem to van Schooten but with an earlier reference (*Organica Conicorum DescriptioThursday, February 25, 2010 1:08:32 PM*, 1646, c. 3, Ex. Math. IV.) and a different solution.

The applet below illustrates the proof from [H. Dorrie, #47]. Point A slides on the X-axis, point B on the Y-axis. The axes meet at the draggable origin O; the endpoints of the axes are also draggable. To see an outline of the curve traced by vertex C, check the box "Trace". If the box "Define triangle" is checked, points A and B move independently so that one can define the shape of triangle ABC. When the box is not checked, moving one of the points A, B causing the other to move too so that the distance between them is preserved.

What if applet does not run? |

Draw a circle L through A, B and O. Let M be the center of L. Line CM intersects L in two points P and Q. Since PQ is a diameter of L,

Think of circle L and points P and Q as being rigidly connected to triangle ABC. The arcs AP and AQ change their location but not their angular magnitude, meaning that P and Q remain on straight lines through O. This allows for shifting a view point. Two points P and Q slide along two perpendicular fixed lines that meet at O. Point C is fixed on line PQ. This is a situation described by the trammel of Archimedes. (So, in particular, since in the right triangle OPQ, OM is the median to the hypotenuse and hence traces a circle with center O.) Thus C describes an ellipse with axes along the orthogonal lines OP and OQ.

Hubert Shutrick made an observation that the center M of circle L too is rigidly connected to triangle ABC. The quadrilateral ACBD is thus rigid too and, since

Hubert further suggested that a good way to draw an ellipse is to take a rod QP of length the sum of the required half axes and one OM with half its length attached with a pivot to the midpoint M of QP so it can fold like a jack knife. Let C be the point on QP such that CP is the length of the minor half axis. A fixed pivot at O allows OM to turn and P is constrained to move along the line that is chosen for the major axis. Then, C describes half of the ellipse while Q wanders up and down the minor axis. You get the other half if you can get Q to pass O. Teachers could get students to do it with bits of cardboard and drawing pins.

### References

- H. Dorrie,
*100 Great Problems of Elementary Mathematics: Their History and Solution*, NewYork, Dover, 1989.

### Conic Sections > Ellipse

- What Is Ellipse?

- Analog device simulation for drawing ellipses
- Angle Bisectors in Ellipse
- Angle Bisectors in Ellipse II
- Between Major and Minor Circles
- Brianchon in Ellipse
- Butterflies in Ellipse
- Concyclic Points of Two Ellipses with Orthogonal Axes
- Conic in Hexagon
- Conjugate Diameters in Ellipse
- Dynamic construction of ellipse and other curves
- Ellipse Between Two Circles
- Ellipse in Arbelos
- Ellipse Touching Sides of Triangle at Midpoints
- Euclidean Construction of Center of Ellipse
- Euclidean Construction of Tangent to Ellipse
- Focal Definition of Ellipse
- Focus and Directrix of Ellipse
- From Foci to a Tangent in Ellipse
- Gergonne in Ellipse
- Pascal in Ellipse
- La Hire's Theorem in Ellipse
- Maximum Perimeter Property of the Incircle
- Optical Property of Ellipse
- Parallel Chords in Ellipse
- Poncelet Porism in Ellipses
- Reflections in Ellipse
- Three Squares and Two Ellipses
- Three Tangents, Three Chords in Ellipse
- Van Schooten's Locus Problem
- Two Circles, Ellipse, and Parallel Lines

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