New Refinements of the Erdös–Mordell Inequality and Barrow’s Inequality

In this paper, two new refinements of the Erdös–Mordell inequality and three new refinements of Barrow’s inequality are established. Some related interesting conjectures are put forward.


Introduction
In 1935, Erdös [1] proposed the following geometric inequality: For any interior point P of the triangle ABC, let R 1 , R 2 , R 3 be the distances from P to the vertices A, B, C, respectively, and let r 1 , r 2 , r 3 be the distances from P to the sides BC, CA, AB, respectively. Then where ∑ denotes the cyclic sums (we shall use this symbol in the sequel). Equality in (1) holds if and only if the triangle ABC is equilateral and P is its center.
Two years later, Mordell and Barrow [2] first proved the inequality (1), and the latter actually obtained the following sharpness: where w 1 , w 2 , w 3 are the lengths of the bisectors of ∠BPC, ∠CPA, ∠APB, respectively.
The above two inequalities have long been famous results in the field of geometric inequalities. The former is called the Erdös-Mordell inequality, which has attracted the interest of many authors and motivated a large number of research papers (see  and the references cited therein).
In 1957, Ozeki [22] first obtained the following generalization of Barrow's inequality (2) for convex polygons: For any interior point P of the convex polygon A 1 A 2 · · · A n , it holds that where R i = PA i and w i denote the lengths of the bisectors of ∠A i PA i+1 (i = 1, 2, · · · , n and A n+1 = A 1 ). Some other discussions about Barrow's inequality and (3) can be found in [4,14,19,21,23,27].
In 2012, when the author considered Oppenheim's inequality (see [24]) the following sharpened version of the Erdös-Mordell inequality was found: with equality if and only if ABC is an isosceles right triangle and P is its circumcenter. Furthermore, by using inequalities (4), (5), and other results, the author obtained a series of refinements for the Erdös-Mordell inequality in [14,16].
In this paper, we shall give two new refinements of the Erdös-Mordell inequality and three new refinements of Barrow's inequality. In addition, we shall present several interesting related conjectures in the last section.

Refinements of the Erdös-Mordell Inequality
In [11], the author proved the following refinement of the Erdös-Mordell inequality: where h a , h b , h c are the corresponding altitudes of the sides BC, CA, AB of the triangle ABC.
Here, we further give the following result: For any interior point P of the triangle ABC, it holds that Equalities in (7) all hold if and only if ABC is equilateral and P is its center.
To prove Theorem 1, we first give several lemmas.

Lemma 1.
For any triangle ABC with sides a, b, c and real numbers x, y, z, it holds that with equality if and only if x : y : For any triangle ABC with sides a, b, c, we have c can be viewed sides of a triangle, and we see that inequality (8) can be obtained by using the following weighted Oppenheim inequality (see [19], p. 681): (where S is the area of ABC) and the following equivalent form of the Heron formula: Remark 1. In the sixth chapter of the monograph [17], the author proved that inequality (8) is equivalent with (9) and the Wolstenholme inequality (52) below.
In the Appendix A of my monograph [17], Theorem A3 gives an equivalent theorem for the geometric transformations, which includes the following conclusion: An inequality involving any interior point P of the triangle ABC, is equivalent to , r 1 , r 2 , r 3 , In fact, this conclusion can be extended to the following: With above notations, the inequality is equivalent to Proof. Let DEF be the pedal triangle of P with respect to the triangle ABC (see Figure 1), and let EF = a p , FD = b p , DE = c p , then it is easy to get Let h 1 , h 2 , h 3 be the distances from P to the side lines EF, FD, DE, respectively, we also easily obtain In addition, by means of the known formula in the triangle ABC (where w a is the bisector of ∠BAC) and the fact that ∠EPF = π − A, we get where w 1 is the bisector of ∠EPF. Two similar relations hold for the bisectors w 2 , w 3 of ∠FPD, ∠DPE, respectively. If we apply inequality (13) to triangle DEF and point P, then Substituting (15), (16), and (18) into this inequality, (14) follows immediately. Conversely, we can obtain (13) from (14) by using the method of proving Theorem A3 in Appendix A of the monograph [17]. Thus, inequality (13) is equivalent with (14). The proof of Lemma 2 is completed.

Lemma 3. For any interior point P of the triangle ABC, it holds that
with equality if and only if r 2 = r 3 .
Proof. Since w 1 ≥ r 1 etc., the second inequality in (7) is evidently valid. In addition, the third inequality of (7) is easily obtained (see [11]). We now prove the first inequality in (7), i.e., By the area formula h a = 2S/a and the identity we see that (20) is equivalent to According to Lemma 2 and the relations (15) and (16), we further know that inequality (22) is equivalent to i.e., But using r 2 r 3 ≤ (r 2 + r 3 ) 2 /4, Lemma 3, and the known formula Thus, in order to prove inequality (23), we only need to prove that Putting x = r 1 /a, y = r 2 /b, z = r 3 /c in inequality (8) of Lemma 1 and noting the fact that we get inequality (25) immediately. Thus, inequality (20) is proved. It is easily known that the equality in (20) holds if and only if ABC is equilateral and P is its center. This completes the proof of Theorem 1.
Now we state and prove the second refinement of the Erdös-Mordell inequality.
Theorem 2. For any interior point P of the triangle ABC, it holds that The first equality in (27) holds if and only if P is the circumcenter of the triangle ABC. The second and third equalities in (27) hold if and only if the triangle ABC is equilateral and P is its center.
Proof. In triangle ABC, we have the following known angle bisector formula: Noting that with equality if and only if b = c. Applying this inequality to BPC, we get Hence, we have that is, Adding ∑ R 2 R 3 to both sides of the above inequality and then squaring root, we obtain Sine w 1 ≥ r 1 etc., the first inequality in (27) obviously holds. Note that the equality in (30) holds if and only if R 2 = R 3 , thus the equality in (31) holds if and only if R 1 = R 2 = R 3 , which means that P is the circumcenter of the triangle ABC. Furthermore, we can conclude that the first equality in (27) holds if and only if P is the circumcenter of the triangle ABC.
Clearly, the second inequality in (27) is equivalent to Removing 2 ∑ r 2 1 to the right and arranging gives the previous Oppenheim inequality (4), which has been proved by the author in different ways (see [12,14]).
For the third inequality in (27), by squaring both sides and arranging, we know that it is equivalent to which was first established by Chu in [30] and proved by the author in another way in [15]. In addition, we have known that both equalities in (4) and (33) hold if and only if ABC is equilateral and P is its center. This completes the proof of Theorem 2.
From Theorem 2, we have Corollary 1. For any interior point P of the triangle ABC, it holds that Furthermore, we can easily obtain the following inequality: Corollary 2. For any interior point P of the triangle ABC, it holds that

Refinements of Barrow's Inequality
In [14], Theorem 4.3 gives the following refinement of the Erdös-Mordell inequality: which is actually equivalent to Now, we point out that for Barrow's inequality (2), the following similar result holds: Theorem 3. For any interior point P of the triangle ABC, it holds that

Equalities in (38) hold if and only if ABC is equilateral and P is its center.
Clearly, the first inequality in (38) is also equivalent to the following interesting form: To prove this inequality, we first prove a strengthening of the previous inequality (5), which is posed by the author in [12] as a conjecture.

Lemma 4. For any interior point P of the triangle ABC, it holds that
with equality if and only if CA = AB and P is the circumcenter of the triangle ABC.
Proof. We let ∠BPC = 2δ 1 , ∠CPA = 2δ 2 , ∠APB = 2δ 3 . By the previous formula (17), we know that inequality (40) is equivalent to to prove the above inequality we only need to prove that (41) Letting √ R 2 = y and √ R 3 = z, (41) then becomes Note that δ 1 , δ 2 , δ 3 can be viewed angles of a non-obtuse triangle. To prove inequality (42), we only need to prove that the following inequality holds for non-obtuse triangles ABC and real numbers y, z: that is, Multiplying both sides by 4(abc) 2 and using the law of cosines, we can transform the proof to the following weighted inequality: If we denote by Q 0 the value of the left-hand side of (45), then it is easy to check the following identity: which shows that inequality Q 0 ≥ 0 holds clearly. Moreover, from (46) we can obtain the following conclusions: (i) if A = π/2, then the equality in (43) holds if and only if yc = zb; (ii) if A < π/2, then the equality in (43) holds if and only if y = z and b = c. According to this conclusion, we can further determine the equality condition of (40), just as mentioned in Lemma 4. This completes the proof of Lemma 4.
We now prove Theorem 3.

Proof.
As the proof of the first inequality (36) given in [14], we can easily prove the first inequality of (38) by using Lemma 4 (we omit the details here). By the power means inequality and Barrow's inequality (2), we have Hence, the second inequality of (38) follows immediately. Moreover, it is easily known that both equalities in (38) hold if and only if ABC is equilateral and P is its center. The proof of Theorem 3 is completed.
Next, we state and prove the second new refinement of Barrow's inequality (2).

Theorem 4.
For any interior point P of the triangle ABC, it holds that The second equality in (47) holds if and only if P is the circumcenter of the triangle ABC.
Proof. Firstly, we prove the first of (47): According to Lemma 2, we only need to prove that Using the law of cosines in triangle EPF and the fact that ∠EPF = π − A (see Figure 1), we have a 2 p = r 2 2 + r 2 3 − 2r 2 r 3 cos ∠EPF = r 2 2 + r 2 3 + 2r 2 r 3 cos A, Thus, we see that inequality (49) is equivalent to But, for any real numbers x, y, z and ABC, we have the following Wolstenholme inequality (see [19]): with equality if and only if x : y : z = sin A : sin B : sin C. Putting x = √ r 1 , y = √ r 2 , z = √ r 3 in (52) and substituting A → (π − A)/2 etc., we get inequality (51) at once. Thus, inequality (48) is proved. The second inequality in (47) follows immediately by adding the previous inequality (30) and its two analogues. Note that the equality in (30) holds if and only if R 2 = R 3 . We conclude that the second equality in (47) holds if and only if R 1 = R 2 = R 3 , which means that the point P is the circumcenter of ABC. The proof of Theorem 4 is completed.

Remark 3.
The author knows that the triangle ABC need not be equilateral when the first equality in (47) holds but does not know what are the barycentric coordinates of P with respect to the triangle ABC.
Now we give an application of Theorem 4. Squaring both sides of the first inequality of (47), we have Then, applying inequality (30), we further get Expanding gives the following: Corollary 3. For any interior point P of the triangle ABC, it holds that In fact, by using the previous inequality (30), we have the following extension: which implies Barrow's inequality (2). Finally, we give the third new refinement of Barrow's inequality: Theorem 5. For any interior point P of the triangle ABC, it holds that The first equality in (55) holds if and only if P is the circumcenter of ABC. The second equality in (55) holds if and only if ABC is equilateral and P is its center.
Proof. In the proof of Theorem 2, we have proved the first inequality in (55). The second inequality in (55) is easily obtained as follows: By (53), we have where the last step used Barrow's inequality (2). It is not difficult to know the equality conditions of inequality chain (55). The proof of Theorem 5 is completed.

Some Open Problems
In this section, we present some interesting conjectures as open problems. For the second inequality in (27), the author guesses that the following refinement is valid.

Conjecture 1.
For any interior point P of the triangle ABC, it holds that A similar conjecture is as follows.

Conjecture 2.
For any interior point P of the triangle ABC, it holds that Remark 4. The last inequality of (57) is actually equivalent to which is Conjecture 2 posed by the author in [15].
Next, we give a reversed inequality similar to the previous inequality (34).

Conjecture 3.
For any interior point P of the triangle ABC, it holds that Considering generalizations of the first inequality of (47), the author presents the following conjecture: Conjecture 4. Let P be an interior point of a convex polygon A 1 A 2 · · · A n (n > 3) and PA i = R i (i = 1, 2, · · · , n), R n+1 = R 1 , A i A i+1 = a i (i = 1, 2, · · · , n, and A n+1 = A 1 ). Then Remark 5. By the previous inequality, (30) we know that the above inequality is stronger than inequality (3).
We have the following refinement of the Erdös-Mordell inequality (see [10]): in which the first inequality can easily be generalized to polygons by applying inequality (30) and w 1 ≥ r 1 . The author believes that the second inequality can also be generalized to polygons as follows: Conjecture 5. Let P be an interior point of convex polygon A 1 A 2 · · · A n (n > 3), and let r i denote the distances from P to the side lines A i A i+1 (i = 1, 2, · · · , n and A n+1 = A 1 ). Then Similarly, we put forward the following conjecture: Conjecture 6. Let P be an interior point of convex polygon A 1 A 2 · · · A n (n > 3), and let w i denote the angle bisectors of ∠A i PA i+1 (i = 1, 2, · · · , n and A n+1 = A 1 ). Then If the above inequality holds, then we can obtain the following refinement of inequality (3): where R i = PA i (i = 1, 2, · · · , n).

Conflicts of Interest:
The author declares that he has no competing interest.