Sharpened versions of the Erdös-Mordell inequality
- Jian Liu^{1}Email author
https://doi.org/10.1186/s13660-015-0716-2
© Liu 2015
Received: 30 October 2014
Accepted: 26 May 2015
Published: 19 June 2015
Abstract
In this paper, we present two sharpened versions of the Erdös-Mordell inequality and extend them to the cases with one parameter. As applications of our results, the Walker inequality and a new inequality in non-obtuse triangles are obtained. We also propose three interesting conjectures as open problems.
Keywords
MSC
1 Introduction
Many authors have given proofs for this inequality by using different tools; see, for example, [2–7]. On the other hand, this inequality has been extended in various directions, we refer the reader to [1, 8–11]. Some other related results can be found in several papers; see [12–20] and references therein.
It is clear that \(R_{1}+\frac{(r_{2}+r_{3})^{2}}{R_{1}}\ge2(r_{2}+r_{3})\) follows from the arithmetic-geometric mean inequality, thus inequality (1.3) implies the Erdös-Mordell inequality (1.1).
Motivated by inequality (1.3), we shall establish in this paper two sharpened versions of the Erdös-Mordell inequality. We shall also extend them to the cases with one parameter.
2 Two results
We state the first main result in the following.
Theorem 1
The Erdös-Mordell inequality (1.1) can easily be obtained from (2.1) and the above-mentioned inequality \(R_{1}+\frac{(r_{2}+r_{3})^{2}}{R_{1}}\ge2(r_{2}+r_{3})\). Therefore, although the value of the left hand of (2.1) is not always greater than or equal to \(2(r_{1}+r_{2}+r_{3})\), inequality (2.1) can still be regarded as a sharpened version of the Erdös-Mordell inequality.
The proof of Theorem 1 needs the following well-known lemma, which will be used in other results of this note.
Lemma 1
We now prove Theorem 1.
Proof
We now consider the equality condition of (2.1). If P lies inside \(\triangle ABC\), then we have strict inequalities \(r_{1}>0\), \(r_{2}>0\), and \(r_{3}>0\). Thus, the equality in (2.5) holds only when \(a=b=c\). Furthermore, by Lemma 1 we conclude that the equality in (2.1) holds if and only if \(\triangle ABC\) is equilateral and P is its center. If P lies on the boundary (except the vertices) of \(\triangle ABC\), then one of \(r_{1}\), \(r_{2}\), \(r_{3}\) is equal to zero. Thus, we deduce that \(\triangle ABC\) must be isosceles when the equality in (2.5) holds. By Lemma 1 we further deduce that the equality in (2.1) holds if and only if \(\triangle ABC\) is a right isosceles triangle and P is its circumcenter. Combining the arguments of the above two cases, we obtain the equality condition of (2.1) as stated in Theorem 1. This completes the proof of Theorem 1. □
Corollary 1
Next, we give a result similar to Theorem 1.
Theorem 2
We now prove Theorem 2.
Proof
Using similar arguments in the proof of Theorem 1, we easily deduce that the equality in (2.10) holds only when the following two cases occur: the \(\triangle ABC\) is equilateral and P is its center or \(\triangle ABC\) is a right isosceles triangle and P is its circumcenter. This completes the proof of Theorem 2. □
Corollary 2
3 Generalizations of Theorem 1 and Theorem 2
In this section, we present generalizations of Theorem 1 and Theorem 2.
Theorem 3
When \(k=0\), then the above theorem reduces to Theorem 1. In order to prove this theorem, we first give the following lemma.
Lemma 2
Proof
Inequality \(Q_{4}\ge0\) can easily be proved. Indeed, \(Q_{4}\) can be viewed a quadratic function of a with positive quadratic coefficient and positive constant term, and its discriminant is given by \(F_{3}=-8bc(b+c)^{2}(b-c)^{2}\). Hence, we have \(Q_{4}\ge0\).
Form the above proofs of \(Q_{i}\ge0\), we easily conclude that the equalities in \(Q_{i}\ge0\) (\(i=1, 2, \ldots, 6\)) are all valid if and only if \(a=b=c\), i.e., \(\triangle ABC\) is equilateral. This completes the proof of Lemma 2. □
In the following, we shall prove Theorem 3. For brevity, we shall, respectively, denote cyclic sums and products over triples \((a, b, c)\), \((r_{1}, r_{2}, r_{3})\), and \((x, y, z)\) by ∑ and ∏.
Proof
When \(k=0\), inequality (3.1) becomes (2.1) and we have obtained the equality conditions (as stated in Theorem 1). When \(k>0\), by Lemma 1 and identity (3.13) we conclude that the equality (3.1) holds if and only if P is the circumcenter of \(ABC\) and the equalities in \(e_{i}\ge0\) (\(i=1, 2, \ldots, 11\)) are all valid. Note that at most one of x, y, z is equal to zero. Thus, the equalities of \(e_{2}\ge0\), \(e_{3}\ge0\), \(e_{4}\ge0\), \(e_{5}\ge0\), \(e_{7}\ge0\), and \(e_{8}\ge0\) occur only when \(a=b=c\). We further deduce that the equality in (3.1) holds if and only if \(\triangle ABC\) is equilateral and P is its center. This completes the proof of Theorem 3. □
We now state and prove the following generalization of Theorem 2.
Theorem 4
Proof
When \(k=1\), inequality (3.14) reduces to (2.10) and we have pointed out the equality conditions in Theorem 2. When \(k>1\), we have \(t>0\) from the assumption. In this case, by Lemma 1 and (3.18) we conclude that the equality in (3.14) holds if and only if P is the circumcenter of \(ABC\) and the equalities in \(m_{i}\ge0 \) (\(i=1,2, \ldots, 6\)) are all valid. Note that at most one of x, y, z is equal to zero. We further deduce that the equality in (3.14) holds if and only if \(\triangle ABC\) is equilateral and P is its center. The proof of Theorem 4 is completed. □
4 Open problems
The author of this paper has found some sharpened versions of the Erdös-Mordell inequality, which have not been proved at present but have been checked by computer. We introduce here three of them as open problems.
A sharpened version of the Erdös-Mordell inequality similar to the inequalities of Theorem 1 and Theorem 2 is as follows.
Conjecture 1
The two conjectured inequalities below are obvious sharpened versions of the Erdös-Mordell inequality.
Conjecture 2
Since we have inequality \(m_{a}\ge w_{a}\) etc., thus (4.2) is stronger than the Erdös-Mordell inequality.
Conjecture 3
From the fact that \(w_{a}\ge h_{a}\) etc., we can see that (4.3) is stronger than the Erdös-Mordell inequality.
Declarations
Acknowledgements
The author would like to thank the referees and the editors for carefully reading the manuscript and making several useful suggestions.
Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.
Authors’ Affiliations
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