Geometric Loci and ChatGPT: Caveat Emptor! †
Abstract
:1. Introduction
ChatGPT and its brethren are constitutionally unable to balance creativity with constraint. They either overgenerate (producing both truths and falsehoods, endorsing ethical and unethical decisions alike) or undergenerate (exhibiting noncommitment to any decisions and indifference to consequences). Given the amorality, faux science and linguistic incompetence of these systems, we can only laugh or cry at their popularity.
With this situation two of the most basic rules of conduct in mathematics are violated: in mathematics, information is passed on free of charge, and everything is laid open for checking.
- If you know how to do it, use computer algebra. If not, do not use it.
- Caveat emptor!
2. Geometric Loci
2.1. Simple Geometric Loci
- Question Consider a circle centered at and passing through . Place a moving point in this circle, C, join it with B and compute the midpoint D of . What is the geometric locus of D?
- ChatGPT Let us denote the center of the circle as , and a point moving on the circle as . Since C lies on the circle passing through , the distance between A and C is equal to the radius of the circle.
- Question Simplify the expression.
- ChatGPT Let us simplify the expression for the coordinates of the midpoint D:
- Question: Eliminate x and y in the equations .
- Question: Give a procedural description in GeoGebra’s input bar of the above construction, using the LocusEquation Command. Use the Execute command in GeoGebra to provide the description.
- ChatGPT: Certainly! Below is a procedural description for the construction using GeoGebra’s input bar. Note that GeoGebra does not have a direct Execute command, but you can input commands directly into the input bar.
- Open GeoGebra: Open the GeoGebra app on your device.
- Create Points A and B: Input in the input bar and press Enter. Input in the input bar and press Enter.
- Create the Circle: Input c = Circle(A, B) in the input bar and press Enter.
- Create a Moving Point C: Input C = PointOn[c] in the input bar and press Enter.
- Join Points B and C: Input = Line(B, C) in the input bar and press Enter.
- Compute Midpoint D: Input D = Midpoint[] in the input bar and press Enter.
- Use the LocusEquation Command: Input LocusEquation() in the input bar and press Enter. This command tells GeoGebra to create a locus equation based on the movement of point C with respect to the midpoint D, and 1 represents the constant sum of distances.
- Animate the Movement of Point C: Right-click on point C. Select “Animation Options”. Choose “Rotate” or “Translate” to make point C move around the circle.
- Observe the Locus: As point C moves around the circle, GeoGebra will dynamically display the locus of the midpoint D.
- in GeoGebra returns the locus graphic object and its equation (Figure 2). There are some mistakes in ChatGPT’s translation: PointOn should be just Point, Midpoint[BC] shoul be Midpoint(B,C), and LocusEquation should only accept one or two arguments. Moreover, GeoGebra indeed has the Execute command.
2.2. The Wallace–Simson Theorem
- into the GeoGebra input bar, we obtain the sought graphic locus and its equation (Figure 3). Note that we could not use the Locus command; since P does not lie on any known object, it is subject to an a posteriori condition.
2.3. Sketchpad Classic Locus Construction
- into GeoGebra, we obtain the locus (shown in red in Figure 4). But this conic is not the searched locus; dragging point G to make line parallel to line a, makes line change, approaching a limit position parallel to and a. Thus, and a do not intersect, I goes to infinity and J is undefined. The point is not in the locus! Things are analogous when dragging point G to make line parallel to line b, thus causing point to be removed from the locus. GeoGebra is unaware of this subtlety and returns the whole ellipse as the locus. GeoGebra locus internals only deal with polynomials, that is, algebraic varieties. In order to correctly describe this locus, it would be necessary to manage different varieties, i.e., locally closed sets (see [22] for an in–depth discussion of this locus).
3. Envelopes of Families of Plane Curves
3.1. The Envelope of Kissing Circles
- in the input bar of the construction, shown in Figure 6, we obtain a long equation in two variables with a degree of 12 in the Algebra window. Factoring it with any computer algebra software or by inputting
- in GeoGebra, we obtain
3.2. The Deltoid of Steiner
4. Conclusions
Author Contributions
Funding
Data Availability Statement
Acknowledgments
Conflicts of Interest
Appendix A. The Wallace–Simson Theorem
- Question Consider a triangle , and a point P. Trace the orthogonal projections from P to the sides of the triangle. Find the locus of point P such that the feet of the orthogonal projections are collinear.
- ChatGPT To find the locus of point P, such that the feet of the orthogonal projections are collinear, let us consider the orthogonal projections of P onto the sides of the triangle .
- Projection onto (the line segment connecting A and B):The equation of line AB is . The foot of the perpendicular from P to will have coordinates .
- Projection onto (the line segment connecting B and C):The equation of line is . The foot of the perpendicular from P to will have coordinates .
- Projection onto (the line segment connecting A and C):The equation of line is . The foot of the perpendicular from P to will have coordinates .
- Question Give a procedural description in GeoGebra of the above construction.
- ChatGPT Certainly! To create the construction in GeoGebra, follow these steps:
- Open GeoGebra: Open GeoGebra on your device or use the online version at www.geogebra.org.
- Create Points: Define the points for vertices A, B, and C:Click on the “Point” tool and create points , , and .
- Define Point P: Click on the “Point” tool again and create a point P with coordinates . You can use the “Input” bar to set the coordinates.
- Create Projections:To represent the projection onto (the x-axis): Click on the “Perpendicular Line” tool. Select point P and line . This creates a perpendicular line from P to .For the projection onto (vertical line at ): Click on the “Perpendicular Line” tool again. Select point P and line .For the projection onto (line ): Click on the “Perpendicular Line” tool once more. Select point P and line .
- Intersect Perpendicular Lines: Create the intersection points between the perpendicular lines and the corresponding sides of the triangle.
- Collinearity Check: Create a line through the three intersection points to check for collinearity.
- Locus Calculation: You can use the “Input” bar or the “Algebra” view to define the locus equation. For the given construction, it is .
- Animate Point P (Optional): To see how the locus changes as P moves, you can use the “Slider” tool or the “Animate” command to vary the coordinates of point P.
- Explore and Analyze: Manipulate the position of point P and observe how the feet of the orthogonal projections move and whether they remain collinear.
Appendix B. Sketchpad Locus
- Question Let and be three fixed points. Let a be the line passing through A and . Let b be the line passing through B and . Let c be the circle with center C and radius 5. Consider a moving point G on the circle c. The line intersects line a in a point I, and line b in a point H. Find the locus of the intersection point J of lines and .
- ChatGPT To find the locus of the intersection point J of lines and , let us first find the equations of lines and c. Then, we can find the equations of lines and in terms of the parameters of the moving point G on circle c. Afterward, we can find the equation of the locus of J.
- Question Give a procedural description suitable for GeoGebra input bar of the construction.
- ChatGPT Sure! Here is a procedural description suitable for GeoGebra’s input bar for constructing the locus of the intersection point J of lines and :
- Define the fixed points , and .
- Define the moving point G on the circle c with center and radius 5:, where t is a parameter.
- Define the coordinates of point .
- Define the coordinates of point .
- Define the equation of line a passing through points A and E:
- Define the equation of line b passing through points B and D:
- Find the intersection point I of line a and :, where satisfies the equation of circle c.
- Find the intersection point H of line b and :, where satisfies the equation of circle c.
- Define the equation of line AH passing through points A and H:
- Define the equation of line passing through points B and I:
- Find the intersection point J of lines and :, where satisfies the system of equations formed by and .
- Locus: Move the parameter t and observe the locus of point J as G moves along the circle c.
Appendix C. The Envelope of Kissing Circles
- Question Compute the envelope of the family of circles with radius 1 centered at a point lying on the circle centered at A(0,0) and radius 1.
- ChatGPT The family of circles with radius 1 centered at points lying on the circle centered at and radius 1 forms a special case known as a “kissing circle" configuration. In this case, each circle is at a tangent to the central circle at a point on its circumference.
Appendix D. The Deltoid of Steiner
- Question Consider a triangle with vertices and . Let D be a point on the circumcircle of the triangle. Compute the envelope of the Simson lines when D moves along its path.
- ChatGPT The Simson line associated with a point D on the circumcircle of a triangle is a line that intersects the sides of the triangle at three points, one on each side. The envelope of Simson lines for all possible positions of D is known as the Simson–Wallace or Simson Envelope.
- Line :
- Line :
- Line :
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ChatGPT | GeoGebra | |
---|---|---|
Example 1 | ✔ (needs user interaction) | ✔ equation, ✔ plot |
Wallace–Simson | × | ✔ equation, ✔ plot |
Sketchpad | × | ∼ equation, ✔ plot |
ChatGPT | GeoGebra | |
---|---|---|
Kissing circles | × | ✔ equation, ✔ plot |
Steiner’s deltoid | × | × equation, × plot |
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Botana, F.; Recio, T. Geometric Loci and ChatGPT: Caveat Emptor! Computation 2024, 12, 30. https://doi.org/10.3390/computation12020030
Botana F, Recio T. Geometric Loci and ChatGPT: Caveat Emptor! Computation. 2024; 12(2):30. https://doi.org/10.3390/computation12020030
Chicago/Turabian StyleBotana, Francisco, and Tomas Recio. 2024. "Geometric Loci and ChatGPT: Caveat Emptor!" Computation 12, no. 2: 30. https://doi.org/10.3390/computation12020030
APA StyleBotana, F., & Recio, T. (2024). Geometric Loci and ChatGPT: Caveat Emptor! Computation, 12(2), 30. https://doi.org/10.3390/computation12020030