Резюме. The computer program “Discoverer”, created by the authors, is the first computer program, able easily to discover new theorems in mathematics, and possibly, the first computer program, able easily to discover new knowledge in science. In this paper we use “Discoverer” in order to investigate the Lalesco products in the geometry of the triangle.

Ключови думи: geometry of the triangle, computer-discovered mathematics, Discoverer.

1. Introduction

The computer program “Discoverer”, created by the authors, is the first computer program, able easily to discover new theorems in mathematics, and possibly, the first computer program, able easily to discover new knowledge in science. See (Grozdev & Dekov, 2014, 2015). In this paper we give an example of the use of the “Discoverer”. We investigate the Lalesco products and we present results obtained by the “Discoverer”.

In this paper the labeling of triangle centers follows ETC (Kimberling). Hence, X(1) denotes the Incenter, \(\mathrm{X}(2)\) denotes the Centroid, etc. Given \(\triangle A B C\), we denote by \(a, b\) and \(c\) the side lengths of \(\triangle A B C, a=B C, b=C A\) a nd \(c=\mathrm{A} B\).

In 2011 Randy Hutson announced the following theorem (See (Kimberling), articles X(99), X(100), X(101), X(1292), X(1293) and X(1296)):

Theorem 1. Let \(I=\{(1,2),(1,3),(1,6),(1,7),(2,6),(3,6),(7,8)\}\). Let \((i, j) \in I\) and let \(L_{1}\) be the reflection of the line \(\mathrm{X}(\mathrm{i}) \mathrm{X}(\mathrm{j})\) in line \(B C\), and define \(L_{2}\) and \(L_{3}\) cyclically. Let \(D=\) \(L_{2} \cap L_{3}, E=L_{3} \cap L_{1}\), , and \(F=L_{1} \cap L_{2}\). The lines \(A D, B E\) and \(C F\) concur in a point denoted by \(\mathrm{LP}(\mathrm{X}(\mathrm{i}), \mathrm{X}(\mathrm{j}))\). The points of concurrence are as follows: \(\mathrm{LP}(\mathrm{X}(1), \mathrm{X}(2))=\mathrm{X}(1293)\), \(\mathrm{LP}(\mathrm{X}(1), \mathrm{X}(3))=\mathrm{X}(100), \mathrm{LP}(\mathrm{X}(1), \mathrm{X}(6))=\mathrm{X}(1292), \mathrm{LP}(\mathrm{X}(1), \mathrm{X}(7))=\mathrm{X}(101)\), \(\mathrm{LP}(\mathrm{X}(2), \mathrm{X}(6))=\mathrm{X}(1296), \mathrm{LP}(\mathrm{X}(3), \mathrm{X}(6))=\mathrm{X}(99)\) and \(\mathrm{LP}(\mathrm{X}(7), \mathrm{X}(8))=\mathrm{X}(1292)\).

Theorem 2. Given \(\triangle A B C\). Let \(P\) and \(U\) be points different from the vertices of \(\triangle A B C\). Let \(L_{1}\) be the reflection of the line \(P U\) in line \(B C\), and define \(L_{2}\) and \(L_{3}\) cyclically. Let \(D=L_{2} \cap L_{3}, E=L_{3} \cap L_{1}\), E = L3ÇL1, and \(F=L_{1} \cap L_{2}\). The lines \(A D, B E\) and \(C F\) concur in a point, which lies on the circumcircle of \(\triangle A B C\).

The construction in theorem 2 is published by T. Lalesco in 1937 and reprinted in 1987. See (Lalesco, 1987).We call the Lalesco product of \(P\) and \(U\) the point of concurrence, defined in theorem 2, and we denote by \(L P(P, U)\) the Lalesco product of \(P\) and \(U\). Note that the Lalesco product is commutative. Also, if point \(V\) lies on the line \(P U\), then \(L P(P, V)=L P(V, U)=L P(P, U)\).

Theorem 3 below is discovered by the computrer program “Discoverer”.

Theorem 3. Given \(\triangle A B C\). Let \(P\) and \(U\) be points different from the vertices of \(\triangle A B C\) and having barycentric coordinates \(P=(p, q, r)\) and \(U=(u, v, w)\). The Lalesco product \(L P(P, U)\) has barycentric coordinates

\((f(a, b, c, p, q, r, u, v, w),(f(b, c, a, q, r, p, v, w, u),(f(c, a, b, r, p, q, w, u, v))\)

where

\[ \begin{aligned} & f(a, b, c, p, q, r, u, v, w)=a^{2}\left(q c^{2} w-b^{2} q w+q u b^{2}+q u c^{2}-a^{2} q u\right. \\ & \left.-a^{2} q w-2 p b^{2} w+b^{2} v r-v p c^{2}+a^{2} v p+a^{2} v r+2 u b^{2} r-v c^{2} r-v p b^{2}\right) \\ & \left(q c^{2} w+a^{2} q w-b^{2} q w+2 q u c^{2}-p b^{2} w-p c^{2} w+a^{2} p w\right. \\ & \left.-a^{2} v r+b^{2} v r+u c^{2} r-a^{2} u r-2 v p c^{2}+u b^{2} r-v c^{2} r\right) \end{aligned} \]

2. Preliminaries

In this section we review some basic facts about barycentric coordinates. For the barycentric coordinates we refer the reader to (Grozdev & Nenkov, 2012a,b).

Given \(\triangle A B C\). The barycentric coordinates with respect to \(\triangle A B C\) are used to define points, lines, circles, triangles, etc. The reference triangle \(A B C\) has vertices \(A=(1,0,0)\), \(B=(0,1,0)\) and \(C=(0,0,1)\). A point is an element of the projective space \(P_{3}(\mathbb{R})\), defined up to a proportionality factor, that is, for \(\forall k \in \mathbb{R} \backslash\{0\}: P=(u, v, w)\) means that \(P \simeq(u, v, w) \simeq(k u, k v, k w)\).

The reflections \(P_{a}, P_{b}\) a and \(P_{c}\) of a point \(P=(p, q, r)\) in the sidelines \(B C, C A, A B\), respectively, are the points (See (Hatzipolakis & Yiu, 2009)):.

\[ \begin{aligned} P_{a} & =\left(-a^{2} p,\left(a^{2}+b^{2}-c^{2}\right) p+a^{2} q,\left(c^{2}+a^{2}-b^{2}\right) p+a^{2} r\right) \\ P_{b} & =\left(\left(a^{2}+b^{2}-c^{2}\right) q+b^{2} p,-b^{2} q,\left(b^{2}+c^{2}-a^{2}\right) q+b^{2} r\right) \\ P_{c} & =\left(\left(c^{2}+a^{2}-b^{2}\right) r+c^{2} p,\left(b^{2}+c^{2}-a^{2}\right) r+c^{2} q,-c^{2} r\right) \end{aligned} \]

The equation of the line joining two points with coordinates \(\left(u_{1}, v_{1}, w_{1}\right)\) and

\(\left(u_{2}, v_{2}, w_{2}\right)\) is

(1) \[ \left|\begin{array}{ccc} u_{1} & v_{1} & w_{1} \\ u_{2} & v_{2} & w_{2} \\ x & y & z \end{array}\right|=0 \]

The intersection of two lines \(L_{1}: p_{1} x+q_{1} y+r_{1} z=0\) and \(L_{2}: p_{2} x+q_{2} y+r_{2} z=0\) is the point

(2) \[ \left(q_{1} r_{2}-q_{2} r_{1}, r_{1} p_{2}-r_{2} p_{1}, p_{1} q_{2}-p_{2} q_{1}\right) \]

Three lines \(p_{i} x+q_{i} y+r_{i} z=0, i=1,2,3\) are concurrent ix + qi y + ri z = 0, i = 1, 2, 3 are concurrent if and only if

(3) \(\left|\begin{array}{lll} p_{1} & q_{1} & r_{1} \\ p_{2} & q_{2} & r_{2} \\ p_{3} & q_{3} & r_{3} \end{array}\right|=0\)

3. Proof of theorem 3.

Proof. Let \(P=(p, q, r)\) and \(U=(u, v, w)\) be p, q, r) and U = (u, v, w) be points distinct from the vertices of

\(\triangle A B C\). D Denote by \(P_{a}, P_{b}\) and \(P_{c}\) the reflections of point \(P\) in the sidelines \(B C, C A\) and \(A B\), respectively. Similarly, define points \(U_{a}, U_{b}\) and \(U_{c}\). By using (1) we obtain that the equation of the line \(L_{1}\) through points \(P_{a}\) a and \(U_{a}\) is \(p_{1} x+q_{1} y+r_{1} z=0\), wher1z = 0, where

\[ p_{1}=a^{2} u r+u b^{2} r-u c^{2} r+v p c^{2}+a^{2} v p-v p b^{2}+a^{2} v r \]

\[ \begin{gathered} -a^{2} p w-p b^{2} w+p c^{2} w-q u c^{2}-a^{2} q u+q u b^{2}-a^{2} q w \\ q_{1}=a^{2}(r u-p w), r_{1}=a^{2}(-u q+v r) . \end{gathered} \]

Similarly we obtain \(L_{2}: p_{2} x+q_{2} y+r_{2} z=0\) where

\[ \begin{gathered} p_{2}=b^{2}(-v r+q w), \\ q_{2}=a^{2} q w+b^{2} q w-q c^{2} w+v p b^{2}+v p c^{2}-a^{2} v p+p b^{2} w \\ -a^{2} v r-b^{2} v r+v c^{2} r-q u b^{2}-q u c^{2}+a^{2} q u-u b^{2} r, \\ r_{2}=b^{2}(-u q+v p), \end{gathered} \] and \(L_{3}: p_{3} x+q_{3} y+r_{3} z=0\), where

\[ \begin{gathered} p_{3}=c^{2}(-v r+q w), q_{3}=c^{2}(r u-p w), \\ r_{3}=q c^{2} w+a^{2} q w-b^{2} q w+u b^{2} r+u c^{2} r-a^{2} u r+q u c^{2} \\ -v c^{2} r-a^{2} v r+b^{2} v r-p b^{2} w-p c^{2} w+a^{2} p w-v p c^{2} . \end{gathered} \]

By using (2), we find the barycentric coordinates of points \(D=L_{\mathrm{B}} \cap L_{\mathrm{C}}, E=L_{\mathrm{C}} \cap L_{\mathrm{A}}\), and \(F=L_{\mathrm{A}} \cap L_{\mathrm{B}}\). Next, we obtain that the equations of the line \(A D\) is \(p_{4} x+q_{4} y+r_{4} z=0\), where

\[ \begin{gathered} p_{4}=0 \\ q_{4}=c^{2}\left(-a^{2} q w+q c^{2} w-b^{2} q w-a^{2} q u+q u c^{2}+q u b^{2}-2 p b^{2} w\right. \\ \left.-v p b^{2}-v c^{2} r+2 u b^{2} r+a^{2} v r+a^{2} v p-v p c^{2}+b^{2} v r\right) \\ r_{4}=b^{2}\left(q c^{2} w+a^{2} q w-b^{2} q w+2 q u c^{2}-p b^{2} w-p c^{2} w+a^{2} p w\right. \\ \left.-a^{2} u r+b^{2} v r-a^{2} v r+u c^{2} r-v c^{2} r-2 v p c^{2}+u b^{2} r\right) \end{gathered} \]

Similarly, we obtain the equations of the lines \(B E\) and \(C F\). By using (3), we prove that the lines \(A D, B E\) and \(C F\) concur in a point. Finally, we find the barycentric coordinates of the point of concurrence \(L P(P, U)\) as the point of concurrence of the lines \(A D\) and \(B E\). The barycentric coordinates of the point \(L P(P, U)\) are given in the statement of theorem 3.

4. Examples of Lalesco products

The computer program “Discoverer” easily produces examples of Lalesco products. The enclosed HTML-files, containing examples of Lalesco products, are produced by the “Discoverer”.

Let \(\Omega\) be the set of all Lalesco products \(L P(P, U)\) where \(P\) is any of the points listed in the enclosed “List 1”, and \(U\) is any of the first ten points, listed in the same list. The enclosed “List K” contains the Lalesco products \(L P(P, U) \in \Omega\) which are available in Kimberling’s ETC, and the rest of the points of \(\Omega\) is listed in the “List D”. Also, the “Discoverer” rewrites the List K to two tables. In our investigation “List K” and “List D” contain 387 and 1268 points, respectively.

Any item of “List K” is a theorem. For example, no 62 of the List K is the following

Theorem 4. The Lalesco Product of X(2) Centroid and X(3) Circumcenter is the Euler reflection point \(X(110)\).

We can easily prove this theorem by using theorem 3. In the barycentric coordinates of the Lalesco product of \(P\) and \(U\), given in theorem 3 we have to substitute the barycentric coordinates of the Centroid for the barycentric coordinates of \(P\), and similarly the barycentric coordinates of the Circumcenter for these of \(U\). Recall that the barycentric coordinates of the Centroid G and the Circumcenter O are as follow: \(G=(1,1,1)\) and \(O=\left(a^{2}\left(b^{2}+c^{2}-a^{2}\right), b^{2}\left(c^{2}+a^{2}-b^{2}\right), c^{2}\left(a^{2}+b^{2}-c^{2}\right)\right)\). After some simplification of the coordinates, we obtain that the fist coordinate of \(L P(\mathrm{X}(2), \mathrm{X}(3))\) is. \(a^{2} /\left(b^{2}-c^{2}\right)\). This is the first b barycentric coordinate of the Euler reflection point, the point \(\mathrm{X}(110)\) in (Kimberling).

5. New remarkable points

The enclosed “List D” contains Lalesco products of remarkable points of the triangle which are not available in the ETC. We may expect that these points are new remarkable points of the triangle. By using theorem 3, we can easily find the barycentric coordinates of these new points. For example, let us consider no.126 in the “List D”:

Theorem 5. The Lalesco Product of X(2) Centroid and X(37) Grinberg Point is not listed in the ETC.

By using theorem 3, we easily find the barycentric coordinates of this new notable point. Recall that the first barycentric coordinate of the Grinberg Point is \(a(b+c)\). The first barycentric coordinate \(u R\) of \(R=L P(\mathrm{X}(2), \mathrm{X}(37))\) is as follows:

\[ \begin{gathered} u R=a^{2}(a-b)(a-c)\left(2 a^{2} b-a^{2} c-2 a b c+2 b^{2} a-b^{2} c-3 c^{3}\right) \\ \left(-2 a^{2} c+a^{2} b+2 a b c-2 a c^{2}+3 b^{3}+c^{2} b\right) \end{gathered} \]

Note that in the “List D” there are points which coincide. In the list “List D” are listed 1268 points, but the list contains only 1109 new notable points of a triangle. These new points are listed in the enclosed “List 2”.

6. Supplementary material

The enclosed file “Lalesco.zip” contains the files quoted in this paper. The reader may download it from the web page of the journal.

REFERENCES

Grozdev, S. & Dekov, D. (2014). Computer-generated mathematics: Points on the Kiepert hyperbola, The Mathmatical Gazette, Vol. 98, no 543, 509 – 511.

Grozdev, S. & Dekov, D. (2015). The Computer Improves the Steiner’s Construction of the Malfatti Circles, Mathematics and Infomatics, vol. 58, no.1, 40 – 51.

Grozdev, S. & Nenkov, V. (2012a). Three remarkable points on the medians of a triangle, Sofia, Archimedes (in Bulgarian).

Grozdev, S. & Nenkov, V. (2012b). On the orthocenter in the plane and in the space, Sofia, Archimedes (in Bulgarian).

Hatzipolakis, A. & Yiu, P. (2009) Reflections in Triangle Geometry, Forum Geometricorum, vol. 9, pp. 301 – 348. http://forumgeom.fau.edu/FG2009volume9/FG200931.pdf

Kimberling, C. Encyclopedia of Triangle Centers, The 5549 points edition of 2013. http://faculty.evansville.edu/ck6/encyclopedia/ETC.html

Lalesco, T. La geometrie du Triangle, Paris Vuibert 1937; Jacques Gabay reprint 1987.