The Complete Strong Version of Blundon’s Inequality

Abstract

In this paper we present some key results related to Blundon’s inequality, its long history, geometric interpretations and implications, as well as highlight some connections to results in other fields of mathematics. We make a case that this is a fundamental inequality in triangle geometry. Also, we provide a new proof for the inequalities (8) and we generalize the strong version of Blundon’s inequalities presented in Theorem 1.

1. Introduction

Many mathematical fields have at least one theorem that is called “fundamental”. Examples are the Fundamental Theorem of Arithmetic, the Fundamental Theorem of Algebra, or the Fundamental Theorem of Calculus. Hence, one may wonder what could make an inequality fundamental? Here we give reasons why we consider that Blundon’s inequality deserves to be called Fundamental Inequality of the Triangle. In support, we provide relevant historical facts, links to other mathematical fields, extensions, as well as an elementary proof of a strong version of Blundon’s inequality and a generalization.

2. Background

In this section we provide some historical background on Blundon’s inequality, highlighting connections to other results, applications, and various extensions.

2.1. Blundon

Professor Jack W. Blundon was born on 20 February 1916 at St. Anthony, Newfoundland, Canada, and was the first head of the Department of Mathematics and Statistics at Memorial University of Newfoundland. He made important contributions to geometry and number theory, many of them published in the American Mathematical Monthly and several other important journals. His 1965 paper “Inequalities associated with the triangle” [1] is a key reference to the geometry of triangle inequalities, together with [2,3]. In his honor, the Canadian Mathematical Society organizes the Blundon Mathematics Contest and the Blundon Seminar Camp for senior high school students, since 1984. Blundon’s work on problem solving has been celebrated through the open book edited since 1994 by Eddy and Parmenter [4], featuring problems in geometry, geometric inequalities, number theory and miscellaneous topics.

2.2. Blundon’s Inequality

Despite being referred to as a single inequality, Blundon’s inequality in fact comprises of two inequalities:

$$\begin{array}{c}\hfill 2R^2+10Rr-r^2-2(R-2r)\sqrt{R^2-2Rr}\le s^2,\end{array}$$

(1)

and

$$\begin{array}{c}\hfill s^2\le 2R^2+10Rr-r^2+2(R-2r)\sqrt{R^2-2Rr}.\end{array}$$

(2)

This result was actually first proved in 1851 by E. Rouché [5], who answered a question of Ramus concerning the necessary and sufficient conditions for three positive real numbers s, R and r to be the semiperimeter, circumradius, and inradius of a triangle.

Over time, these inequalities have been rediscovered in various equivalent forms. A standard simple proof was given by Blundon [1] and it is based on the following algebraic property of the roots of a cubic equation.

Proposition 1.

The roots $x_1$, $x_2$ and $x_3$ of the cubic equation

$$\begin{array}{c}\hfill x^3+a_1{x}^2+a_{2}x+a_3=0,\end{array}$$

are the side lengths of a (nondegenerate) triangle if and only if the following conditions hold:

(i) 

$18a_1{a}_2{a}_3+a_1^2{a}_2^2-27a_3^3-4a_2^3-4a_1^3{a}_3>0$;

(ii) 

$-a_1>0, a_2>0, -a_3>0$;

(iii) 

$a_1^3-4a_1{a}_2+8a_3>0$.

For more historical details one can consult the monograph of Mitrinović, Pečarić and Volenec [6]. In 1971, Bottema [7] discussed Blundon’s inequality, calling this old result the fundamental inequality of the triangle, while authors use the name Blundon’s fundamental inequality for inequalities (1) and (2). Blundon’s inequality has many applications in Euclidean geometry, particularly in the field of geometric inequalities.

2.3. Geometric Interpretation

Bottema [7] obtained a geometric interpretation of Blundon’s inequality, rewriting the fundamental inequalities (1) and (2) as

$$\begin{array}{c}\hfill {(r^2+s^2)}^2+12R{r}^3-20Rr{s}^2+48R^2{r}^2-4R^2{s}^2+64R^{3}r\le 0.\end{array}$$

(3)

D. Andrica and C. Barbu [8] gave a direct geometric proof to Blundon’s inequalities, while the proof and some comments are presented in [9]. In the notations of Figure 1, where O is the circumcenter, I is the incenter and N is Nagel’s point of a triangle $ABC$, they applied the Law of Cosines in the triangle $ION$, to deduce the closed formula

$$\begin{array}{c}\hfill cos\widehat{ION}={\displaystyle \frac{2R^2+10Rr-r^2-s^2}{2(R-2r)\sqrt{R^2-2Rr}}}.\end{array}$$

(4)

Since $-1\le cos\widehat{ION}\le 1$, Formula (4) directly implies Blundon’s inequalities (1) and (2), revealing their geometric nature.

Extending these results, other Blundon-type inequalities were obtained in [10], using the same idea with different triplets instead of I, O, N.

If $\varphi$ denotes $min\{\left|A-B\right|,\left|B-C\right|,\left|C-A\right|\}$, where A, B, C are the angles of $ABC$, then the following improvement to inequalities (1) and (2) was proposed in [11]:

$$\begin{array}{c}\hfill -cos\varphi \le cos\widehat{ION}\le cos\varphi ,\end{array}$$

(5)

with alternative geometric proof provided in [12].

A geometric interpretation of Blundon’s inequality is given by S. Wu and Y. Chu [13], by constructing two isosceles triangles $A_1{B}_1{C}_1$ and $A_2{B}_2{C}_2$ inscribed in the same circle of radius R, such that the vertex angles have the measures: $A_1=2arcsin\left({\displaystyle \frac{1}{2}}+{\displaystyle \frac{1}{2}}\sqrt{1-{\displaystyle \frac{2r}{R}}}\right)$ and $A_2=2arcsin\left({\displaystyle \frac{1}{2}}-{\displaystyle \frac{1}{2}}\sqrt{1-{\displaystyle \frac{2r}{R}}}\right)$, respectively (see also the triangles $A_{min}{B}_{min}{C}_{min}$ and $A_{max}{B}_{max}{C}_{max}$ in Section 4).

3. What Makes an Inequality Fundamental?

Inspired by Polya’s statement [14]: “Mathematics is governed by inequality; equality is rather a particular case”, it is clear that inequalities play an important role in geometry. There are thousands of inequalities, structured in many categories: algebraic inequalities (arithmetic–geometric mean inequality is a prominent example), geometric inequalities (inequalities for triangles, quadrilaterals, etc.), isoperimetric inequalities in the plane, matrix inequalities [15], or numerous named inequalities (Hölder’s inequality [16], Minkowski’s inequality [17], Chebyshev’s inequality [18], Jensen’s inequality [19], etc.).

Geometric inequalities have been explored since ancient times, with classical examples being given in the Euclid’s first book of Elements:

1.

In any triangle, the greater side subtends the greater angle;

2.

In any triangle, the greater angle is subtended by the greater side;

3.

In any triangle, the sum of any two sides (taken in any possible way) is greater than the remaining side.

Back to Blundon’s inequalities, these are fundamental in triangle geometry because they represent the necessary and sufficient conditions for the existence of a triangle with given elements circumradius R, inradius r, and semiperimeter s.

The famous Romanian mathematician and poet I. Barbu (Dan Barbilian) [20] asserts: “Somewhere in the high realm of geometry there is a bright spot where it meets poetry.” From this perspective, one can say that Blundon’s inequality has the beauty of a poem, connecting algebra, mathematical analysis, Euclidean geometry and hyperbolic geometry.

3.1. Links to Algebra

The algebraic character of inequalities (1) and (2) is discussed in references [21,22] and an elementary proof to the weak form of (1) is given in [6]. Other results connected to (1) are contained in [11].

Furthermore, L. Euler himself has studied the existence problem for a triangle with given $\{s,r,R\}$ and he presented a cubic equation (see [21,22]) from which we can obtain the fundamental triangle inequality, as follows:

The side length of a any triangle satisfies the cubic equation:

$$\begin{array}{c}\hfill x^3-2s{x}^2+(s^2+r^2+4Rr)x-4sRr=0.\end{array}$$

G. Dospinescu, M. Lascu, C. Pohoaţă and M. Tetiva [23] proposed a new algebraic demonstration to the weaker Blundon’s inequality

$$\begin{array}{c}\hfill s\le 2R+(3\sqrt{3}-4)r.\end{array}$$

This inequality is a direct consequence of the fundamental triangle inequality.

3.2. Consequences and Applications

The importance of Blundon’s inequality is also highlighted by the large number of mathematicians who applied it in their research, or found extensions and generalisations of this result over a long period of time. We provide here some illustrative examples.

An equivalent form to Rouché–Blundon’s inequality was proposed by Sondat [24] in 1850 and solved by Lemoine [25] in 1891 as follows: The necessary and sufficient condition for the existence of a triangle $ABC$ with the elements $R,r,s$ is

$$\begin{array}{c}\hfill s^4-2(2R^2+10Rr-r^2)s^2+r{(4R+r)}^3\le 0,\end{array}$$

with equality if and only if the triangle $ABC$ is isosceles.

M. Bataille [26] discovered a beautiful consequence of Blundon’s inequality:

$$\begin{array}{c}\hfill s^2\ge {\displaystyle \frac{2r(r+4R)(2R-r)}{R}}.\end{array}$$

Using Blundon’s inequalities we also obtain Kooi’s inequality [27]

$$\begin{array}{c}\hfill s^2\le {\displaystyle \frac{R{(4R+r)}^2}{2(2R-r)}},\end{array}$$

which is equivalent to Garfunkel–Bankoff’s inequality [28]:

$$\begin{array}{c}\hfill {tan}^2{\displaystyle \frac{A}{2}}+{tan}^2{\displaystyle \frac{B}{2}}+{tan}^2{\displaystyle \frac{C}{2}}\ge 2-8sin{\displaystyle \frac{A}{2}}sin{\displaystyle \frac{B}{2}}sin{\displaystyle \frac{C}{2}}.\end{array}$$

Furthermore, Gerretsen–Stening’s inequalities [6]

$$\begin{array}{c}\hfill 16Rr-5r^2\le s^2\le 4R^2+4Rr+r^2,\end{array}$$

follow very simply from the fundamental triangle inequality.

Finally, another famous inequality resulting from Blundon’s inequality is Emmriche’s inequality [6]:

$$\begin{array}{c}\hfill r\le (\sqrt{2}-1)R,\end{array}$$

for which a short geometric proof (valid for nonacute triangles) was given in [29].

3.3. Links Between Geometry and Mathematical Analysis

An interesting connection between geometry and mathematical analysis was suggested by R. Satnoianu in [30,31], who used analytical results to prove Blundon’s inequality and some of its consequences. For the proof he first introduces The principle of the isosceles triangle which states that: Assume that $ABC$ is a triangle which $0\le C\le B\le A\le \pi ,$ and let $g=g(A,B,C)$ be a real-valued differentiable function in

$$\begin{array}{c}\hfill T=\left\{(A,B,C)\left|A,B,C\ge 0,A+B+C=\pi \right.\right\}\subset {\mathbb{R}}^3.\end{array}$$

Let $T^i$ be the subset of T formed by all isosceles triangles. Suppose that $g\ge 0$ for all triangles in $T^i$ and there is at least one isosceles triangle $T_0^i$ on which g attains a global minimum in T. Then $g(A,B,C)\ge 0$ in T.

In [32] R. Satnoianu extends the fundamental triangle inequality, as:

$$\begin{array}{c}\hfill a^n+b^n+c^n\le 2^{n+1}{R}^n+2^n\left(3^{1+{\displaystyle \frac{n}{2}}}-2^{n+1}\right)r^n,\mathrm{for}\mathrm{any}n\ge 0,\end{array}$$

where $a,b,c$ are the side lengths of the triangle $ABC$.

A Blundon-type inequality for the fundamental triangle inequality in hyperbolic geometry was obtained by Svrtan and Veljan [33], while Andrica, Barbu, and Minculete [10] proved the relations (1) and (2) with barycentric coordinates, providing a geometric approach for generating Blundon-type inequalities.

4. The Complete Strong Version of Blundon’s Inequality

Andrica, Barbu, and Pişcoran introduced the Blundon configuration [29]. Denote by $T(R,r)$ the family of all triangles inscribed in the circle of radius R and center O (fixed), and circumscribed to the circle of radius r and center I (fixed). Inequalities (1) and (2) give the exact interval containing the semiperimeter s for triangles in family $T(R,r)$, in terms of R and r. The triangles in the family $T(R,r)$ are situated “between” two extremal triangles $A_{min}{B}_{min}{C}_{min}$ and $A_{max}{B}_{max}{C}_{max}$ determined by

$$s_{min}=\sqrt{2R^2+10Rr-r^2-2(R-2r)\sqrt{R^2-2Rr}}$$

(6)

and

$$s_{max}=\sqrt{2R^2+10Rr-r^2+2(R-2r)\sqrt{R^2-2Rr}}.$$

(7)

These triangles are isosceles with respect to the vertices $A_{min}$ and $A_{max}$. The key elements of the Blundon configuration are represented in Figure 1.

The above computations show that a triangle in the family $T(R,r)$ is perfectly determined up to a congruence by the angle A. In this way, we obtain the distribution of triangles in the family $T(R,r)$ as illustrated in Figure 2.

The strong version of Blundon’s inequality was given in [29]: In the Blundon’s configuration, the function $A\mapsto s\left(A\right)$ denoting the semiperimeter of triangle $ABC$, is strictly decreasing on the arc $\widehat{A_{max}{B}_{min}}$; that is, we have the inequalities

$$s\left(A_{max}\right)\ge s\left(A\right)\ge s\left(B_{min}\right).$$

(8)

Here we provide a new proof for the inequalities (8) and we generalize the strong version of Blundon’s inequalities. In what follows, the following identity is useful:

$$\begin{array}{c}\hfill s={\displaystyle \frac{r+atan\frac{A}{2}}{tan\frac{A}{2}}}={\displaystyle \frac{r+2RsinAtan\frac{A}{2}}{tan\frac{A}{2}}}={\displaystyle \frac{r}{tan\frac{A}{2}}}+2RsinA={\displaystyle \frac{r}{tan\frac{A}{2}}}+4R{\displaystyle \frac{tan\frac{A}{2}}{1+{tan}^2\frac{A}{2}}}.\end{array}$$

Denoting $t=tan{\displaystyle \frac{A}{2}}$, $t\in (0,\infty )$, one can define the function function $f:(0,\infty )\to \mathbb{R}$,

$$\begin{array}{c}\hfill f\left(t\right)={\displaystyle \frac{r}{t}}+4R{\displaystyle \frac{t}{1+t^2}},\end{array}$$

whose derivative is given by

$$\begin{array}{c}\hfill f^{\prime }\left(t\right)={\displaystyle \frac{-(4R+r)t^4+2(2R-r)t^2-r}{t^2{(1+t^2)}^2}}.\end{array}$$

The equation $f^{\prime }\left(t\right)=0$ has two positive solutions

$$\begin{array}{c}\hfill t_1=\sqrt{{\displaystyle \frac{2R-r-2\sqrt{R^2-2Rr}}{4R+r}}}\mathrm{and}{t}_2=\sqrt{{\displaystyle \frac{2R-r+2\sqrt{R^2-2Rr}}{4R+r}}}.\end{array}$$

From the sign of $f^{\prime }$, function f is strictly decreasing on the intervals $\left(0,t_1\right)$ and $\left(t_2,\infty \right)$, and strictly increasing on $\left(t_1,t_2\right)$. The following results capture the geometric essence of the values $t_1$ and $t_2$. Denoting $a_m=B_{min}{C}_{min}$ and $a_M=B_{max}{C}_{max}$, we have [29]:

$$\begin{array}{cc}\hfill a_m^2& =4r\left(2R-r+2\sqrt{R^2-2Rr}\right)\hfill \end{array}$$

(9)

$$\begin{array}{cc}\hfill a_M^2& =4r\left(2R-r-2\sqrt{R^2-2Rr}\right).\hfill \end{array}$$

(10)

Lemma 1.

The following relations hold:

$$\begin{array}{c}\hfill t_1=tan{\displaystyle \frac{B_{min}}{2}}\mathit{and}{t}_2=tan{\displaystyle \frac{C_{max}}{2}}.\end{array}$$

(11)

Proof. 

According to Figure 1 and using Formula (9), we have

$$\begin{array}{cc}\hfill {tan}^2{\displaystyle \frac{B_{min}}{2}}& ={\displaystyle \frac{4r^2}{a_m^2}}={\displaystyle \frac{4r^2}{4r\left(2R-r+2\sqrt{R^2-2Rr}\right)}}={\displaystyle \frac{r}{2R-r+2\sqrt{R^2-2Rr}}}\hfill \\ & ={\displaystyle \frac{r\left(2R-r-2\sqrt{R^2-2Rr}\right)}{{\left(2R-r\right)}^2-4\left(R^2-2Rr\right)}}={\displaystyle \frac{2R-r-2\sqrt{R^2-2Rr}}{4R+r}}=t_1^2,\hfill \end{array}$$

and the first relation follows. Similarly, from Figure 1 and Formula (10) we obtain:

$$\begin{array}{cc}\hfill {tan}^2{\displaystyle \frac{C_{max}}{2}}& ={\displaystyle \frac{4r^2}{a_M^2}}={\displaystyle \frac{4r^2}{4r\left(2R-r-2\sqrt{R^2-2Rr}\right)}}={\displaystyle \frac{r}{2R-r-2\sqrt{R^2-2Rr}}}\hfill \\ \hfill & ={\displaystyle \frac{r(2R-r+2\sqrt{R^2-2Rr})}{{\left(2R-r\right)}^2-4\left(R^2-2Rr\right)}}={\displaystyle \frac{2R-r+2\sqrt{R^2-2Rr}}{4R+r}}=t_2^2,\hfill \end{array}$$

and we get the second expression. □

Theorem 1

(The complete strong version of Blundon’s inequality). In the Blundon configuration depicted in Figure 2, the function $A\mapsto s\left(A\right)$ (where $s\left(A\right)$ denotes the semiperimeter of the triangle $ABC$), is strictly decreasing on the arc $\widehat{A_{max}{B}_{min}}$, strictly increasing on $\widehat{B_{min}{C}_{max}}$, and strictly decreasing on $\widehat{C_{max}{A}_{min}}$.

Proof. 

We first prove the result for the arc $\widehat{A_{max}{B}_{min}}$. By Lemma 1, for $A_1$ and $A_2$ with

$$A_{max}\le A_1<A_2\le B_{min},$$

we have

$$\begin{array}{c}\hfill 0<tan{\displaystyle \frac{A_{max}}{2}}\le tan{\displaystyle \frac{A_1}{2}}<tan{\displaystyle \frac{A_2}{2}}\le tan{\displaystyle \frac{B_{min}}{2}}=t_1.\end{array}$$

From the variation of function f, we obtain

$$\begin{array}{c}\hfill f(tan{\displaystyle \frac{A_{max}}{2}})\ge f(tan{\displaystyle \frac{A_1}{2}})>f(tan{\displaystyle \frac{A_2}{2}})\ge f(tan{\displaystyle \frac{B_{min}}{2}}),\end{array}$$

hence $s\left(A_{max}\right)\ge s\left(A_1\right)>s\left(A_2\right)\ge s\left(B_{min}\right)$.

We can proceed in a similar way for the arcs $\widehat{B_{min}{C}_{max}}$ and $\widehat{C_{max}{A}_{min}}$. □

5. Conclusions

The beauty and wide applicability of these inequalities, the bridges they create between algebra, mathematical analysis, Euclidean geometry and hyperbolic geometry are good reasons on why Blundon’s inequalities deserve the name of the fundamental inequality of triangle. This topic is far from being closed, and the complete strong version of Blundon’s inequality proposed here will surely generate future developments.