Образователни технологии
THE COMPUTER IMPROVES THE STEINER’S CONSTRUCTION OF THE MALFATTI CIRCLES
Резюме. The computer program “Discoverer”, created by the authors, is the first computer program, which is able easily to discover new theorems in Mathematics, and possibly, the first computer program,which is able easily to discover new knowledge in science. In this paper we give a detailed description of an improvement of the classical Steiner’s solution of the construction of the Malfatti circles, discovered by the computer program “Discoverer”. We use the theory of the complexity of the geometric constructions in order to obtain a numerical measure of the complexity of the solutions.
Ключови думи: computer-generated mathematics, Euclidean geometry, Discoverer, Malfatti circles, Steiner’s solution.
1. Introduction
The computer program “Discoverer” is the first computer program, which is able easily to discover new theorems in mathematics, and possibly, the first computer program, which is able easily to discover new knowledge in science, see (Grozdev & Dekov, 2013, 2014a,b,c).
In this paper we give a detailed description of an improvement of the Steiner’s solution of the Malfatti circles construction, created by the “Discoverer” (Grozdev & Dekov, 2013). We use the theory of the complexity of the geometric constructions in order to obtain a numerical measure of the complexity of the solutions.
The construction of the Malfatti circles is one of the famous mathematical problems. The problem was posed by the Italian geometer Gian Francesco Malfatti in 1803. A simple construction of the Malfatti circles with a compass and a ruler has been published by the great Swiss geometer Jacob Steiner in 18261 (Tabov & Lazarov, 1990). As far as the authors know, the improvement of the Steiner’s construction of the Malfatti circles, discovered by the”Discoverer”, is the first essential improvement of an important result in Mathematics, discovered by a computer, and possibly, the first improvement of an important result in science, discovered by a computer.
2. Complexity of geometric constructions
The first measure of the complexity of geometric constructions is proposed by Lemoine1 (Tabov & Lazarov, 1990). In this paper we use the measure of Lazarov and Tabov (Tabov & Lazarov, 1990) which is summarized in Table 1. This measure specifies the Lemoine’s measure. The explanation of row 1 in Table 1 is as follows. To place the edge of the ruler in coincidence with a point (Lemoine’s operation R1) – one point. To place the edge of the ruler in coincidence with a second point – one point. To draw a straight line (Lemoine’s operation R2) – one point. Hence, we obtain 3 points for drawing a straight line. The explanation of rows 2 and 3 in Table 1 is similar.
Table 1
We use aslo the Fransois Labelle’s measure2. The Labelle’s measure is as follows: the complexity of a construction is defined to be the number of drawing operations (lines and circles) that are performed.
In the header cells in the tables below “LT” means “Lazarov-Tabov complexity” and “L” means “Labelle’s complexity”.
We illustrate the measures with four examples. We will use these examples in the next sections.
Example 1. See Figure 1. Construct an internal angle bisector of a given angle. The edge of the angle is labeled A, and the rays of the angle are labeled R1 and R2. The construction and the complexity are given in Table 2.
Table 2
Figure 1
Example 2. See Figure 2. Construct the projection of a point A on a line L. See Table 3.
Table 3
Example 3. See Figure 3. Construct the reflection of a point A in a line L. See Table 4.
Table 4
Figure 2
Figure 3
Example 4. See Figure 4. Given circle c, line L tangent to circle c and a second line L1. Construct the tangent to c, different from L, through the point which is the intersection of the lines L and L1. See Table 5. Below “ L2 = second tangent (c, L, L1)” means that L2 is the second tangent to c, constructed as in Table 5. We use this notation below in Table 6, rows 18 and 19.
Table 5
Figure 4
3. Steiner’s solution
The Malfatti problem is as follows: Within a given triangle draw three circles each of which is tangent to the other two and to two sides of the triangle.
Given ABC, that is, given points A, B and C and lines BC, CA and AB. The Steiner’s construction has the following stages:
Stage 1. Construct the internal angle bisectors and the incenter of ABC.
Stage 2. Construct the vertices of the de Villiers triangle.
Stage 3. Construct the Malfatti-Steiner point.
Stage 4. Construct the Malfatti central triangle.
Stage 5. Construct the Malfatti circles.
The steps of the construction are given in Table 6. In Table 6, the internal angle bisectors, labeled by L1, L2 and L3, are constructed at steps 1, 2 and 3, and the incenter, labeled by I, is constructed at step 4. The vertices of the de Villiers triangle, labeled by V1, V2 and V3, are constructed at steps 7, 10 and 15. The Malfatti-Steiner point, labeled by S, is constructed at step 20. The vertices of the Malfatti central triangle, labeled by O1, O2 and O3, are constructed at steps 24, 26 and 28. The Malfatti circles, labeled by c1, c2 and c3, are constructed at steps 30, 32 and 34. See Figures 5-9.
Table 6
We see that the complexity of the Steiner’s solution is 290, if we use the LazarovTabov measure, and the complexity is 74, if we use the Labelle’s measure.
Figure 5. Stage 1 of the Steiner’s construction.
Figure 6. Stage 2 of the Steiner’s construction.
Figure 7. Stage 3 of the Steiner’s construction.
Figure 8. Stage 3 of the Steiner’s construction. Closer look.
Figure 9. Stage 4 of the Steiner’s construction.
Figure 10. Stage 5 of the Steiner’s contruction.
4. Replacements
A replacement of stage 3 of the Steiner’s solution is given by Richard Guy3 (Guy, 2007). The Guy’s construction of stage 3 is given in Table 7. See Figure 11.
Table 7
Figure 11. The Guy’s construction of stage 3
A replacement of stage 4 of the Steiner’s solution is given by the computer program “Discoverer” (Grozdev & Dekov, 2013). The construction of “Discoverer” of stage 4 is given in Table 8. See Figure 12.
Table 8
Figure 12. The Discoverer’s construction of stage 4
Table 9 gives comparison of the variants of the Steiner’s construction.
Table 9
5. Conclusion
The Lazarov-Tabov measure of stage 4 of the Steiner’s construction is 50, while the improved by “Discoverer” stage 4 has measure 12. The Labelle’s measure of stage 4 of Steiner’s construction is 12, while the improved by “Discoverer” stage 4 has measure 3. The complexity of the improved by “Discoverer” stage 4 is in the first case 24%, and in the second case 25% of the complexity of the Steiner’s stage 4. Hence, the computer program “Discoverer” has discovered an essential improvement of stage 4 of the Steiner’s construction of the Malfatti circles.
NOTES
1. Geometrography, http://en.wikipedia.org/wiki/
2. Labelle, F., The complexity of geometric constructions, http://www.cs.mcgill. ca/~sqrt/cons/constructions.html
3. Malfatti circles, http://en.wikipedia.org/wiki/
REFERENCES
Grozdev, S., Dekov, D. (2013). Towards the first computer-generated encyclopedia (Bulgarian), Mathematics and Informatics, 56 (1), 49-59.
Grozdev, S., Dekov, D. (2014a). Computer-generated mathematics: Kosnita products in Euclidean geometry (Bulgarian), Mathematics and Informatics, 57 (4), 355-363.
Grozdev, S., Dekov, D. (2014b). Computer-Generated Mathematics: A New Relation between the Steiner Circumellipse and the Kiepert Hyperbola, Journal of ComputerGenerated Mathematics, 9 (4), http://www.ddekov.eu/j/
Grozdev, S., Dekov, D. (2014c). Computer-generated mathematics: Points on the Kiepert hyperbola, The Mathmatical Gazette, November 2014.
Guy, Richard K. (2007). The lighthouse theorem, Morley & Malfatti - a budget of paradoxes, American Mathematical Monthly, 114 (2), 97–141.
Tabov, J., Lazarov, B. (1990). Geometric Constructions (in Bulgarian), Sofia: Narodna Prosveta.