# Discovering Geometric Inequalities: The Concourse of GeoGebra Discovery, Dynamic Coloring and Maple Tools

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## Abstract

**:**

## 1. Introduction

**Proposition**

**1.**

`Maple`to definitely settle Proposition 1. Finally, in Section 7, we include some conclusions and propose lines of research to improve the performance and versatility of GeoGebra in future versions.

## 2. The GeoGebra Automated Reasoning Tools

**Remark**

**1.**

- (i)
- The use of complex versus real geometry tools: Points in the Euclidean plane have real coordinates, while the algebraic algorithms used within this framework are designed for algebraically closed fields. Therefore, when we state that $V\left(H\right)\subset V\left(T\right)$ we are including points in $V\left(H\right)$ with complex coordinates, which is more than what we actually need. We could be missing valid theses which hold for all real points in $V\left(H\right)$, but not on the whole complex variety (see [17,18]). For example, let us consider the following statement, usually called Clough’s conjecture [19], namely, the equality of
- -
- the sum of segments $l=EC$, $m=FB$, and $n=GA$, where $E,F,G$ are the feet of the perpendiculars to the sides of an equilateral triangle from an arbitrary point D,
- -
- and $3p/2$, where p is the length of side $AB$, i.e., $3p/2$ is half the perimeter $3p$ of the equilateral triangle $ABC$.

See [20] for detailed synthetic and algebraic proofs. Now, the algorithms we have implemented in GeoGebra, if asked about the`Relation`between $l+m+n$ and $3/2\xb7p$, proceed, first, checking the numerically approximate equality of both quantities and, if it holds, starting a symbolic protocol by considering the ideal generated by all the polynomial equations describing the hypotheses. We emphasize here the terms polynomial equations, because all the lengths $l,m,n,p$ are described by expressing ${l}^{2},{m}^{2},{n}^{2},{p}^{2}$ as the sum of the squares of the differences of the coordinates of the extremes of the corresponding segments, say, ${l}^{2}={({c}_{1}-{e}_{1})}^{2}+{({c}_{2}-{e}_{2})}^{2}$, etc. There are neither square roots nor choice of positive signs for defining $l,m,n,p$, as this approach would require real algebraic geometry methods. Thus, our hypotheses ideal includes much more cases (in the complex geometry context) than intuitively expected, and the thesis $l+m+n=3/2\xb7p$, holds only on some of them (for example, on those cases where $l,m,n,p$ are positive).Accordingly, GeoGebra Discovery declares that the formulated statement is “true on parts, false on parts”, see Figure 2. This means that the underlying algorithm detects, without having to compute a primary decomposition of the involved hypotheses ideal, that the statement holds true on some components and is false on some others, see item iii) below.On the other hand, asking directly for the proof of Clough’s statement, see Figure 3, an affirmative answer is obtained, together with a list (too long in this case to be fully displayed in the Figure) of construction instances that should be avoided for the truth of the statement: degenerate cases such as when the triangle collapses and, also, variants of the thesis statement that hold just on the primary components of the hypotheses where the given thesis is false.This is related to the possibility, in this case, to modify the thesis so that it collects all possible sign choices for $l,m,n,p$, by multiplying $(l+m+n)-3/2\xb7p$ by all possible variants of such formula, changing signs for $l,m,n,p$, such as $\left(\right(l-m+n)-3/2\xb7p)$, $\left(\right(l-m+n)+3/2\xb7p)$, etc. See [21] for a detailed study of this way of understanding real statements in a complex geometry context.Of course, the reason for choosing a complex algebraic geometry approach for GeoGebra automated reasoning tools is because algorithms based on complex geometry are generally much more efficient than their real counterparts, and this aspect is of special relevance for the implementation of GARTs in software with educational purposes. - (ii)
- Non-degeneracy conditions: In Geometry statements, whenever we mention objects such as line segments, polygons, circles..., we implicitly assume certain conditions on the points that are used to define those figures and render a theorem as true. As an example, in a theorem on triangles we always assume that its vertices are non-collinear points of the plane. These conditions are not easy to interpret by the GATPs and require special care when dealing with them. The problem of non-degeneracy conditions, which depends upon the choice of independent variables in the construction, the way the construction itself is carried out, or the interpretation we can give to what a degenerate construction is, has been studied extensively in the literature and the reader is referred to [12,22,23,24,25] or [26] to gain a better understanding of the intricacies surrounding it.
- (iii)
- True on parts, false on parts statements: As already explained in item i), quite often the hypothesis variety that is obtained from the algebraization process (via complex geometry algorithms which avoid the computationally costly decomposition of the hypothesis variety into irreducible components) can have irreducible, non-degenerate components of maximal dimension, such that the thesis holds on some of the components, while not on others. This situation was thoroughly treated in [27], and gave rise to a new category of true on parts, false on parts geometric statements.

**Strategy**

**1.**

**Strategy**

**2.**

**Strategy**

**3.**

`Relation(<list of objects>):`This higher level command performs two distinct operations: In the first place it proceeds to make a numerical comparison of the objects, and outputs an answer concerning geometric properties such as equality/inequality, parallelism, collinearity, concurrency, etc. Together with this numerical output, the user finds a “More...” button that, when clicked, gives way to a deeper, symbolic analysis of the numerical result previously obtained, resorting to the commands that mention below and proving or rejecting its truthfulness. See Figure 2.`Prove(<Boolean Expression>):`This command is the basic symbolic automatic theorem prover in GeoGebra, and its output can be (generally)`true`, (generally)`false`or`undefined/?`(when GeoGebra cannot decide the question). In order to introduce easily the Boolean expressions of geometric nature, GeoGebra offers a number of different commands such as`AreParallel(<line>,<line>)`,`AreCollinear(<Point>,<Point>, <Point>)`,`AreConcyclic(<Point>,<Point>,<Point>,<Point>)`, etc.`ProveDetails(<Boolean Expression>):`Improved version of the previous command, offering as output more information on degeneracy conditions whenever they are easy to translate into a simple statement. See Figure 3.`LocusEquation(<Arguments>):`This command offers the possibility of finding the algebraic equation for a geometric locus, obtained also by symbolic means. Besides, the command shows on the graphic window the curve corresponding to the equation. Among the several options for choosing the arguments, one which will be useful later to our purposes is`LocusEquation(<Boolean expression>, <Point>)`; this command will find the equation for the locus of points satisfying the given Boolean expression.`Discover(<Point>):`This command, one of the latest additions in GeoGebra Discovery, provides as output from a geometric construction a whole set of geometric statements concerning properties about the lines, segments and other geometric elements related to the point provided as argument. The inner workings of the command follow the idea exposed in Strategies 1 and 3 above.

## 3. New Tools in GeoGebra Discovery: Handling Geometric Inequalities

- If j is odd, then$$\begin{array}{c}{S}_{j}=\{({u}_{1},\dots ,{u}_{n}):({u}_{1},\dots ,{u}_{n-1})\in S\phantom{\rule{4pt}{0ex}}\hfill \\ \hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\wedge \phantom{\rule{4pt}{0ex}}{f}_{(j-1)/2}({u}_{1},\dots ,{u}_{n-1})<{u}_{n}<{f}_{(j+1)/2}({u}_{1},\dots ,{u}_{n-1})\}.\hfill \end{array}$$
- If j is even, then$${S}_{j}=\{({u}_{1},\dots ,{u}_{n}):({u}_{1},\dots ,{u}_{n-1})\in S\phantom{\rule{4pt}{0ex}}\wedge \phantom{\rule{4pt}{0ex}}{u}_{n}={f}_{j/2}({u}_{1},\dots ,{u}_{n-1})\}.$$

- For $n=1$, a CAD of $\mathbb{R}$ is a finite partition $\mathbb{R}={\u2a06}_{j=1}^{2k+1}{S}_{j}$ of the real line in k points and the $k+1$ open intervals which constitute their complement (it can be viewed as a stack over a point). A point is called a 0-cell and an open interval a 1-cell of the CAD.
- For $n>1$, a CAD of ${\mathbb{R}}^{n}$ is a partition of ${\mathbb{R}}^{n}={\u2a06}_{i=1}^{r}({T}_{i}\times \mathbb{R})={\u2a06}_{i=1}^{r}\left({\u2a06}_{j=1}^{2{k}_{i}+1}{S}_{ij}\right)$ in stacks ${T}_{i}\times \mathbb{R}={\u2a06}_{j=1}^{2{k}_{i}+1}{S}_{ij}$, whose elements are semialgebraic sets, and such that the collection $\left\{{T}_{i}\right\}$, $i=1,\dots ,r$ is a CAD of ${\mathbb{R}}^{n-1}$ (here we have renamed and reindexed the elements of the $(n-1)$—dimensional CAD to avoid accumulation of subindices). If ${T}_{i}$ is a k-cell, then ${S}_{ij}$ is either a $(k+1)$-cell (if j is odd) or a k-cell (if j is even). Besides, ${\pi}_{n-1}\left({S}_{ij}\right)={T}_{i}$ and, by the recursive nature of the construction, the collection of sets $\left\{{\pi}_{k}\left({S}_{ij}\right)\right\}$ (with repeated elements) forms a CAD of ${\mathbb{R}}^{k}$.

`Mathematica`,

`Maple`,

`Reduce`or

`QEPCAD`, and GeoGebra Discovery, in its most recent versions, has incorporated them through its realgeom extension, see [38]. In its most recent release, GeoGebra Discovery embeds the Tarski/QEPCAD B program (see [39,40,41]) to perform these CADs, opening the possibility of proving geometric inequalities without resorting to external programs. The commands in GeoGebra that put in action this new set of tools are the following:

`Compare(<Expression, Expression>):`This command performs a sequence of tasks that can be briefly described as follows (see [40]):- The first two points of the construction are identified with (0,0) and (1,0). This reduces the number of variables required, and avoids some degenerate configurations.
- By using only the equalities defining the hypothesis, there is a first trial to check for an equality relation between the input expressions. This is handled by the Giac CAS embedded in GeoGebra.
- If equality relations are not found, the set of equations in the hypothesis is processed, eliminating unnecessary variables and generating a quantified formula.
- Now, via the Tarski/QEPCAD B software, this quantified formula is converted into a quantifier-free formula, by means of a CAD algorithm that outputs an inequality for the quotient of the input expressions.
- This inequality is translated and displayed into a simple form, easy to understand by the user.

See Figure 5 for an example of the performance of the`Compare`command regarding the statement of Proposition 1.`Relation(<list of objects>):`As mentioned before, this higher level command performs two distinct operations ([28]): first, proceeds to make a numerical comparison of the objects, and outputs an answer concerning geometric properties, including—this is the more recent improvement—inequalities, etc. Then the user finds a “More...” button that, if clicked, starts a deeper, symbolic analysis of the numerical result previously obtained, proving or rejecting its truthfulness. A detailed example is developed in Section 5, and shown in Figures 10 and 11.

## 4. Visualization of Real Curves and the Dynamic Color Scanning Method

**Example**

**1.**

`ImplicitCurve((x-1)^2+y^2),`

`ImplicitCurve((x-1)^4+y^4),`

`LocusEquation()`, since it makes use of

`ImplicitCurve()`to represent loci graphically.

**Example**

**2.**

`Maple`face similar problems with its plotting commands. The graphic output of the set of instructions

`with(plots);`

`implicitplot(x^2+y^2-x^3, x=-1..1, y=-1..1);`

`implicitplot`command uses a sample-based method which is not able to detect accurately isolated points. However,

`Maple`includes other representation options that allow a more precise output for the case of real algebraic curves, by means of the package

`algcurves`. Executing now

`with(algcurves);`

`plot_real_curve(x^2+y^2-x^3, x, y);`

**Problem**

**1.**

`ImplicitCurve`command in GeoGebra, see Figure 9a. Now, in order to exemplify how to visualize this locus through our dynamic color scanning method approach, let us start associating some colors to point $P(x,y)$, as follows. First, we remark that GeoGebra allows to set independently dynamic values of the three color channels RGB that colorize an object in GeoGebra. The numeric value for each channel ranges between 0 (no color) and 1 (full color), and for other values z outside this range GeoGebra applies a periodic pattern given by

- Red: c/s
- Green: 1-abs(s-c)
- Blue: s/c,

`c = Distance(P,C)`and

`s = Distance(P,A)+Distance(P,B)`. Observe that we will get for the point P the white color only when $c=s$, since values assigned to red and blue channels are positive and mutually inverse, and all three channels will give odd numbers (in fact, 1) only in this case.

- The periodic patterns that frequently arise in form of concentric rings of similar colors are due to the periodic behavior of the dynamic colors when the values introduced in the RGB channels move outside the $[0,1]$ range, as previously mentioned (see Figure 9b).
- A way to restrict the values for the dynamic color channels in the range $[0,1]$ is obtained by using exponential expressions of the form
`Red: e^(-abs(f(P)))``Green: e^(-abs(g(P)))``Blue: e^(-abs(h(P))),`

so that only when simultaneously the functions $f,g,h$, depending on point P, are equal to 0 the white color will be assigned to P. In our example, by assigning to the color channels the values`Red: e^(-abs(1-c/s))``Green: e^(-abs(s-c))``Blue: e^(-abs(1-s/c)),`

we obtain Figure 9c.Besides, if we substitute in a given RGB channel, for instance the red one, the expression in the exponent $\mathrm{abs}\left(f\right(P\left)\right)$ by $\left(\mathrm{abs}\right(k\phantom{\rule{4pt}{0ex}}f\left(P\right))-\mathrm{floor}(k\phantom{\rule{4pt}{0ex}}\mathrm{abs}\left(f\right(P\left)\right))$, where k is a convenient positive constant, we get the mantissa of $k\phantom{\rule{4pt}{0ex}}\mathrm{abs}\left(f\right(P\left)\right)$, which drops abruptly to zero when $f\left(P\right)$ crosses integer values, producing distinct lines because of the sudden color change in the red channel. This trick leads to the appearance of level curves connecting points P with the same $f\left(P\right)$ value, which will be more densely distributed by choosing a greater value for the constant k. We will make use of this technique later in Section 6.1. - Since we have three different channels that can hold independent values, by playing wisely with them, we have at our disposal up to three different properties to visualize in our scanning method. An illustration of how to take advantage of this feature is shown in Figure 8b, where an investigation on the Fermat’s Point is carried out (see the section on “Escáner del Punto Fermat” in [44], for details on the construction).

- Set a rectangular region to be scanned. Let us say it has as upper left corner the point $({x}_{0},{y}_{0})$, and dimensions $w\times h$.
- Shrink the size of the radius for the point P to its minimum size.
- Create a slider t that will be used as the x-coordinate of P, and runs from ${x}_{0}$ to ${x}_{0}+w$.
- Set the coordinates of P to $(t,{y}_{0})$ and rename the point to $A1$. This trick allows us to automatically assign the point P to the cell $A1$ of the GeoGebra spreadsheet. Now we can create a list of points $Ai$, $1\le i\le n$, by taking advantage of the spreadsheet features, starting first by adding the point $A2=A1-(0,\u03f5)$, with the same properties as A1, and dragging downwards (as it is usually done when working with spreadsheets) the cell A2. This column of points (all of them inheriting the properties we set up for $A1$) translates in the graphic window into a scanning vertical segment, which will sweep the region as the slider t runs through its defined range. A recommended value for $\u03f5$ is $0.02$ in order to obtain a good quality of the scanning image. Therefore, for covering our rectangular region of height h we should create around $n\approx 50$ h points.

## 5. A Case Study: Relation between the Perimeter and Circumradius of a Triangle

`Relation($a+b+c$, R)`and we obtain (see Figure 10) an initial, numerical output indicating that the quantities $a+b+c$ and R do not coincide. Not very informative indeed, but if we press the “More...” button (see Figure 11) the realgeom extension enters in action and quickly shows that the relation $a+b+c\le 3\sqrt{3}\phantom{\rule{0.277778em}{0ex}}R$ holds.

`LocusEquation(<Boolean Condition>, <Point>)`becomes handy, since if we ask GeoGebra to provide us with the geometric locus of the points in the plane satisfying the equality relation $a+b+c=3\sqrt{3}\phantom{\rule{0.277778em}{0ex}}R$ we should obtain an algebraic curve giving us information on the set of points we are looking for. Were Proposition 1 correct, this locus should reduce to the points ${C}_{1}({\textstyle \frac{1}{2}},{\textstyle \frac{\sqrt{3}}{2}})$ and ${C}_{2}({\textstyle \frac{1}{2}},{\textstyle \frac{\sqrt{3}}{2}})$, which are the only ones producing equilateral triangles together with A and B.

`LocusEquation($a+b+c==3\sqrt{3}\phantom{\rule{0.277778em}{0ex}}R$, C)`, obtaining the following equation for this geometric locus:

## 6. Combining Visual and Symbolic Strategies to Overcome Difficulties

#### 6.1. The Visual Approach Using the Dynamic Color Scanning Method

`LocusEquation()`command, let us approach the problem from a more visual strategy, by using the dynamic color scanning method described in Section 4. Let us choose as scanning area the rectangle R with two opposite vertices located at $(-2,-2)$ and $(3,2)$, which contains points A, B of our construction. We have to assign dynamic values to the three RGB color channels of the scanning points, and these values should depend on some kind of measure of the “distance” of the point to the curve $P(x,y)=0$. As an initial choice, we could just choose the values $({e}^{-\left|P\right(x,y\left)\right|},{e}^{-\left|P\right(x,y\left)\right|},{e}^{-\left|P\right(x,y\left)\right|})$ for each point $(x,y)$, but the high values that the polynomial P attains on the considered region (for instance, $P(2,2)=448988778496$) produces undesirable effects in the visualization process, so that it is better to adjust these values in order to obtain a better display. A good choice to get a neat output is obtained by assigning to each point $Ci$ of the scanning segment the following values (see Figure 13 and Figure 14):

- Red: e^(-(abs(k Bi)-floor(abs(k Bi))))
- Green: e^(-abs(Bi))
- Blue: e^(-(abs(Bi)-floor(abs(Bi)))),

**Conjecture**

**1.**

`Maple`and its algebraic toolkit as auxiliary assistant.

#### 6.2. The Symbolic Approach Using Maple Commands

`Maple`and a variety of its available tools to perform a thorough algebraic study of the situation in order to completely elucidate the set of real points in $P(x,y)=0$.

`Maple`to quickly settle down Proposition 1. To achieve this, we need the

`RegularChains`package and its commands to perform CADs:

- >
- with(RegularChains):with(ChainTools):with(SemiAlgebraicSetTools):R:=PolynomialRing([y,x])cad:=CylindricalAlgebraicDecompose(P,R,method = recursive,output=piecewise)

`Maple`is capable of completing these tasks related to working with polynomials in real coefficients, in the sequel, we proceed to give a more guided, step-by-step treatment of the problem, by using lower level

`Maple`commands that can also be translated to other software packages such as the Xcas/Giac environment.

`Maple`. We define now the ideal containing our hypothesis, which we denote by Hypo:

- >
- restart:with(PolynomialIdeals):Hypo:=PolynomialIdeal(u-(1/2),(x-1)*(u-((x+1)/2))+y*(v-(y/2)),h^2-(x-u)^2-(y-v)^2,a^2-(x-1)^2-y^2,b^2-x^2-y^2,c^2-1, variables={x,y,u,v,h,a,b,c});

- >
- HilbertDimension(Hypo);EliminationIdeal(Hypo,{x,y});

- >
- T:=PolynomialIdeal(u-(1/2),(x-1)*(u-((x+1)/2))+y*(v-(y/2)),h^2-(x-u)^2-(y-v)^2,a^2-(x-1)^2-y^2,b^2-x^2-y^2,c^2-1,r^2-3,a+b+c-3*r*h,variables={x,y,u,v,h,a,b,c,r});HilbertDimension(T);

- >
- P:=EliminationIdeal(T,{x,y});

`Maple`also includes graphic tools to plot real algebraic curves, in the same spirit as we tried with GeoGebra we can see whether

`Maple`outputs (see Figure 16) successfully the real set of points for this curve:

- >
- with(algcurves): plot_real_curve(P(x,y),x,y);

**Step****1:**- Check that $P(x,y)=0$ for $x\in \{0,{\textstyle \frac{1}{2}},1\}$ has real solutions in y, and determine them all.
**Step****2:**- Check that $P(x,y)=0$ for $x\notin \{0,{\textstyle \frac{1}{2}},1\}$ has not real solutions in y.

**Step****1:**- We first set $x=0$ and compute $P(0,y)$, factorizing the result.

- >
- simplify(subs(x=0,P(x,y))); factor(%);

- >
- sturmseq(297*y^4 + 432*y^3 + 658*y^2 + 432*y + 297,y);sturm(%,y, -infinity, infinity);

- >
- simplify(subs(x=1,P(x,y)));

- >
- simplify(subs(x=1/2,P(x,y)));

- >
- sturmseq(1936*y^4 + 744*y^2 + 81,y);sturm(%,y, -infinity, infinity);$$\begin{array}{ccc}& \left[{y}^{4}+\frac{93}{242}{y}^{2}+\frac{81}{1936},{y}^{3}+\frac{93}{484}y,-{y}^{2}-\frac{27}{124},y,1\right]& \\ & 0\end{array}$$
- >
- sturmseq(432*y^4 + 152*y^2 + 27,y);sturm(%,y, -infinity, infinity);$$\begin{array}{ccc}& \left[{y}^{4}+\frac{19}{54}{y}^{2}+\frac{1}{16},{y}^{3}+\frac{19}{108}y,-{y}^{2}-\frac{27}{76},y,1\right]& \\ & 0\end{array}$$

**Step****1:**- We start by considering again the polynomial $P(x,y)$, but now displayed as a polynomial in $\left(\mathbb{R}\right[x\left]\right)\left[y\right]$, that is, as a polynomial in the variable y with coefficients in the domain $\mathbb{R}\left[x\right]$:

- >
- P(y):=collect(P(x,y),y);

- >
- F(x):=discrim(P(y),y);

- >
- factor(F_1);factor(F_2);$${(x-1)}^{8}$$$$\begin{array}{ccc}& & (221709312{x}^{10}-1108546560{x}^{9}+5547202304{x}^{8}-15537529856{x}^{7}+23360155968{x}^{6}\hfill \\ & & -20355008960{x}^{5}+10731606224{x}^{4}-3448222560{x}^{3}+658900764{x}^{2}-70266636x\hfill \\ & & +3485889){(4{x}^{2}+23)}^{2}{(4{x}^{2}-8x+27)}^{2}{(2x-1)}^{6}{(x-1)}^{32}\hfill \end{array}$$

- >
- sturmseq(221709312*x^10 - 1108546560*x^9 + 5547202304*x^8−15537529856*x^7 + 23360155968*x^6 - 20355008960*x^5+10731606224*x^4 - 3448222560*x^3 + 658900764*x^2−70266636*x + 3485889,x);sturm(%,x,-infinity, infinity);$$\begin{array}{ccc}& & [{x}^{10}-5{x}^{9}+\frac{21668759}{866052}{x}^{8}-\frac{15173369}{216513}{x}^{7}+\frac{121667479}{1154736}{x}^{6}-\frac{28913365}{314928}{x}^{5}\hfill \\ & & +\frac{670725389}{13856832}{x}^{4}-\frac{11972995}{769824}{x}^{3}+\frac{6100933}{2052864}{x}^{2}-\frac{59147}{186624}x+\frac{11737}{746496},{x}^{9}-\frac{9}{2}{x}^{8}\hfill \\ & & +\frac{21668759}{1082565}{x}^{7}-\frac{106213583}{2165130}{x}^{6}+\frac{121667479}{1924560}{x}^{5}-\frac{28913365}{629856}{x}^{4}+\frac{670725389}{34642080}{x}^{3}\hfill \\ & & -\frac{2394599}{513216}{x}^{2}+\frac{6100933}{10264320}x-\frac{59147}{1866240},-{x}^{8}+4{x}^{7}-\frac{152575271}{23851348}{x}^{6}+\frac{123806941}{23851348}{x}^{5}\hfill \\ & & -\frac{105485273}{47702696}{x}^{4}+\frac{5223331}{11925674}{x}^{3}-\frac{6179517}{381621568}{x}^{2}-\frac{1656315}{381621568}x+\frac{68607}{1526486272},\hfill \\ & & -{x}^{7}+\frac{7}{2}{x}^{6}-\frac{6032902658003487}{1200060777167108}{x}^{5}+\frac{9163449689593045}{2400121554334216}{x}^{4}-\frac{7901689078731841}{4800243108668432}{x}^{3}\hfill \\ & & +\frac{3852119358162855}{9600486217336864}{x}^{2}-\frac{246472468139943}{4800243108668432}x+\frac{52410697894269}{19200972434673728},-{x}^{6}+3{x}^{5}\hfill \\ & & -\frac{1923113575493489063067}{544181504190477281099}{x}^{4}+\frac{1125319630034591720639}{544181504190477281099}{x}^{3}\hfill \\ & & -\frac{2699431145391007808109}{4353452033523818248792}{x}^{2}+\frac{374878642014550049949}{4353452033523818248792}x\hfill \\ & & -\frac{129123207926862901389}{34827616268190545990336},-{x}^{5}+\frac{5}{2}{x}^{4}-\frac{6120084968285261526685869}{5247608252952953776114636}{x}^{3}\hfill \\ & & -\frac{7877786359908984300515573}{10495216505905907552229272}{x}^{2}+\frac{297440786451944785973148517}{440799093248048117193629424}x\hfill \\ & & -\frac{113685261871877652900325591}{881598186496096234387258848},{x}^{4}-2{x}^{3}+\frac{83459519425974095718301535996311}{55852716316393303993363363418712}{x}^{2}\hfill \\ & & -\frac{27606803109580791724938172577599}{55852716316393303993363363418712}x+\frac{6814525055230976067861308675045}{111705432632786607986726726837424},\hfill \\ & & -{x}^{3}+\frac{3}{2}{x}^{2}-\frac{264697672012697346416360283393844501}{357631291633235791535031047332757050}x\hfill \\ & & +\frac{21470506549019862662211189931866494}{178815645816617895767515523666378525},-{x}^{2}+x\hfill \\ & & -\frac{27892930064141325504042418797332472427}{11908197738863253638405327183317787758},x-\frac{1}{2},-1]\hfill \end{array}$$$$\begin{array}{cc}& 0\end{array}$$

**Proposition**

**2.**

- >
- subs(x=-1,P(y)); sturmseq(%,y):sturm(%,y,-infinity, infinity);$$\begin{array}{c}88209{y}^{16}+2128788{y}^{14}+22521286{y}^{12}+130093668{y}^{10}+434474577{y}^{8}\hfill \\ \hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}+846168336{y}^{6}+944574048{y}^{4}+559312128{y}^{2}+136048896\hfill \end{array}$$$$\begin{array}{cc}& 0\end{array}$$
- >
- subs(x=1/3,P(y)); sturmseq(%,y):sturm(%,y,-infinity,infinity);$$\begin{array}{c}88209{y}^{16}-9612{y}^{14}-41114{y}^{12}-892{y}^{10}+\frac{293419}{27}{y}^{8}\hfill \\ \hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}+\frac{1093808}{243}{y}^{6}+\frac{630112}{729}{y}^{4}+\frac{20224}{243}{y}^{2}+\frac{256}{81}\hfill \end{array}$$$$\begin{array}{cc}& 0\end{array}$$
- >
- subs(x=2/3,P(y)); sturmseq(%,y):sturm(%,y,-infinity,infinity);$$\begin{array}{c}88209{y}^{16}-9612{y}^{14}-41114{y}^{12}-892{y}^{10}+\frac{293419}{27}{y}^{8}\hfill \\ \hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}+\frac{1093808}{243}{y}^{6}+\frac{630112}{729}{y}^{4}+\frac{20224}{243}{y}^{2}+\frac{256}{81}\hfill \end{array}$$$$\begin{array}{cc}& 0\end{array}$$
- >
- subs(x=2,P(y)); sturmseq(%,y):sturm(%,y,-infinity,infinity);$$\begin{array}{c}88209{y}^{16}+2128788{y}^{14}+22521286{y}^{12}+130093668{y}^{10}+434474577{y}^{8}\hfill \\ \hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}+846168336{y}^{6}+944574048{y}^{4}+559312128{y}^{2}+136048896\hfill \end{array}$$$$\begin{array}{cc}& 0\end{array}$$

## 7. Conclusions and Further Research

`Maple`.

- (i)
- perform straightforwardly a dynamic color scanning, avoiding the setup process which is needed at present and improving its performance;
- (ii)
- utilize the Giac and Tarski/QEPCAD embedded systems to increase the power of the GART’s tools present in GeoGebra, rendering unnecessary to make use of external software to visualize and study certain geometric problems (see for instance [53]) which can involve equalities or inequalities in their formulation.

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## Abbreviations

CAD | Cylindric Algebraic Decomposition |

CAS | Computer Algebra System |

DGS | Dynamic Geometry Software |

GART | Geometry Automated Reasoning Tool |

GATP | Geometry Automated Theorem Prover |

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**Figure 1.**Proving with GeoGebra that the distances from a point P in the perpendicular bisector of segment $AB$ to its extremes are equal.

**Figure 3.**(

**a**) Asking GeoGebra Discovery to prove that the sum $l+m+n$ is equal to the semiperimeter $3/2f$. (

**b**) GeoGebra Discovery replies that the statement is true except in some degenerate cases (described in the list after true in the last line of the Algebra window).

**Figure 5.**Discovering the inequality between the perimeter and the radius of the circumcenter of a triangle.

**Figure 6.**GeoGebra output for the implicit curves ${(x-1)}^{2}+{y}^{2}=0$ (success, subfigure (

**a**)) and ${(x-1)}^{4}+{y}^{4}=0$ (failure, subfigure (

**b**)).

**Figure 7.**Graphic output of

`Maple`for the curve ${x}^{2}+{y}^{2}-{x}^{3}=0$ (

**a**) with the

`implicitplot()`command; (

**b**) with the

`plot_real_curve()`command.

**Figure 8.**(

**a**) The classic Mandelbrot set, created with the dynamic color scanning method. (

**b**) An application to investigate the Fermat point of an arbitrary triangle.

**Figure 9.**Locus of P such that $d(P,A)+d(P,B)=d(P,C)$, displayed (

**a**) with the traditional GeoGebra tools, (

**b**,

**c**) with the dynamic color scanning method.

**Figure 13.**(

**a**) Setting up the dynamic color scanner. (

**b**,

**c**) Basic parameters of the scanning points. (

**d**) Parameter values for the RGB channels.

Construction Element | Variables | Polynomial Equation |
---|---|---|

Points $A(0,0)$, $B(0,0)$, $C(x,y)$ | $\{x,y\}$ | |

Perpendicular Bisector l to $AB$ | $\left\{u\right\}$ | $u-{\textstyle \frac{1}{2}}=0$ |

Perpendicular Bisector m to $BC$ | $\{x,y,u,v\}$ | $(x-1)(u-{\textstyle \frac{x+1}{2}})+y(v-{\textstyle \frac{y}{2}})=0$ |

Intersection of l and m is $D(u,v)$ | $\{u,v\}$ | |

Segment $h=CD$ | $\{x,y,u,v,h\}$ | ${h}^{2}-{(x-u)}^{2}-{(y-v)}^{2}=0$ |

Segment $a=BC$ | $\{x,y,a\}$ | ${a}^{2}-{(x-1)}^{2}-{y}^{2}=0$ |

Segment $b=CA$ | $\{x,y,b\}$ | ${b}^{2}-{x}^{2}-{y}^{2}=0$ |

Segment $c=AB$ | $\{x,y,c\}$ | ${c}^{2}-1=0$ |

Square Root of 3 | $\left\{r\right\}$ | ${r}^{2}-3=0$ |

Locus Equation | $\{a,b,c,h,r\}$ | $a+b+c-3\xb7r\xb7h$=0 |

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**MDPI and ACS Style**

Recio, T.; Losada, R.; Kovács, Z.; Ueno, C.
Discovering Geometric Inequalities: The Concourse of GeoGebra Discovery, Dynamic Coloring and Maple Tools. *Mathematics* **2021**, *9*, 2548.
https://doi.org/10.3390/math9202548

**AMA Style**

Recio T, Losada R, Kovács Z, Ueno C.
Discovering Geometric Inequalities: The Concourse of GeoGebra Discovery, Dynamic Coloring and Maple Tools. *Mathematics*. 2021; 9(20):2548.
https://doi.org/10.3390/math9202548

**Chicago/Turabian Style**

Recio, Tomás, Rafael Losada, Zoltán Kovács, and Carlos Ueno.
2021. "Discovering Geometric Inequalities: The Concourse of GeoGebra Discovery, Dynamic Coloring and Maple Tools" *Mathematics* 9, no. 20: 2548.
https://doi.org/10.3390/math9202548