Two geometric inequalities involved two triangles
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1 OCTOGON MATHEMATICAL MAGAZINE Vol. 17, No.1, April 009, pp ISSN , ISBN , Two geometric inequalities involved two triangles Yu-Lin Wu 16 ABSTRACT. In this short note, we prove two geometric inequality conjectures involved two triangles posed by Liu [9]. 1. INTRODUCTION AND MAIN RESULTS For ABC, let a, b, c be the side-lengths, the area, s the semi-perimeter, R the circumradius and r the inradius, respectively. Denote by w a, w b and w c the interior bisectors of angles, and h a, h b, h c the altitudes, respectively. It s has been a long time since the scholar studied the inequality involved two triangles. The very famous one is the following Neuberg Pedoe s inequality [11]. a (b + c a + b (c + a b + c (a + b c 16 Recently, Chinese scholar studied some geometric inequalities involved two triangles. For example, Zhang [14] and Gao [4] proved the inequalities as follows, respectively. a a + b b + c c 16 a (b + c a + b (c + a b + c (a + b c 48 An [1] obtained several inequalities as follows. (aa aa + bb + cc 4 bb + bb cc + cc aa 16 Received: Mathematics Subject Classification. 51M16. Key words and phrases. Geometric Inequality; Triangle.
2 194 Yu-Lin Wu 1 aa bb + 1 bb cc + 1 cc aa a (s a + b (s b + c (s c 4 Wu [1] proved the following two inequalities ( sin A h a h a + h b h b + h ch c 3 4 (aa + bb + cc 1 h a h a + 1 h b h + 1 b h c h c 1 aa + bb + cc Leng [8] showed the proof of the inequality as follows. sin A w a w a + w b w b + w cw c 3 4 (aa + bb + cc Jiang [6] proved the following inequality. sin A + a sin b + sin c 1 ( 1 R + 1 r J. Liu [9] posed the following two interesting geometric inequality conjectures in 008. Conjecture 1. For ABC and A B C, prove or disprove (b + ccot A + (c + acot + (a + bcot 4(w a + w b + w c. (1.1 Conjecture. For ABC and A B C, and real numbers x, y, z, prove or disprove x aa + y bb + z cc 4 3 (yzw aw a + zxw b w b + xyw cw c. (1. We prove the two conjectures in this paper.
3 Two geometric inequalities involved two triangles 195. PRELIMINARY RESULTS Lemma 1. ([3, 5] For real numbers x 1, x, x 3, y 1, y, y 3 such that x 1 x + x x 3 + x 3 x 1 0 and y 1 y + y y 3 + y 3 y 1 0, the following inequality holds. (y + y 3 x 1 + (y 3 + y 1 x + (y 1 + y x 3 (x 1 x + x x 3 + x 3 x 1 (y 1 y + y y 3 + y 3 y 1 (.1 With equality holds if and only if x 1 Lemma. ([] In ABC, we have y 1 = x y = x 3 y 3. cot A cot B cot C 3 3. (. Lemma 3. (10, 13 In ABC, we have w a w b + w b w c + w c w a 3r(4R + r (.3 Lemma 4. In ABC, we have w a + w b + w c 3 ab + bc + ca. (.4 Proof. With well-known inequalities [] w a s(s a, etc. We have wa + wb + w c s(s a + s(s b + s(s c = s. (.5 From inequality (.5 and Lemma 3, we obtain (w a + w b + w c s + 4Rr + 6r. (.6 With known identity ab + bc + ca = s + 4Rr + r, we get 9 4 (ab + bc + ca (s + 4Rr + 6r = 5 4 [s 16Rr + 5r + 4r(R r]. (.7
4 196 Yu-Lin Wu From identity (.7, Gerretssen s inequality s 16Rr 5r and Euler s inequality R r, we can conclude that 9 4 (ab + bc + ca (s + 4Rr + 6r 0 s + 4Rr + 6r 3 ab + bc + ca. (.8 Inequality (.4 follows from inequality (.6 and (.8 immediately. Thus, we complete the proof of Lemma 4. Lemma 5. (Wolstenholme s inequality, see [7] For ABC and real numbers x, y, z, we have x + y + z yz cos A + zxcos B + xy cos C, (.9 with equality holds if and only if x : y : z = sin A : sinb : sin C. By Lemma 1, we get 3. THE PROOF OF CONJECTURE 1 (b + ccot A + (c + acot + (a + bcot (ab + bc + ca ( cot A A cot + cot cot + cot cot (3.1 By Lemma and AM GM inequality, we obtain cot A cot + cot cot + cot cot A 3 (cot A 3 cot cot = 9. (3. With inequality (3.1-(3., together with Lemma 4, we can conclude that inequality (1.1 holds. The proof of conjecture 1 is complete.
5 Two geometric inequalities involved two triangles THE PROOF OF CONJECTURE With known inequalities w a bc cos A, etc, we get 4 3 (yz bcb c cos 4 3 (yzw aw a + zxw b w b + xyw cw c A cos A + zx cac a cos B cos + xy aba b cos C cos = = 4 3 [yz bcb c (cos A + A + cos A A + zx cac a (cos B + + cos B + xy aba b (cos C + + cos C ] For A+A + B+ + C+ = π, then by Lemma 5, we have + (4.1 yz bcb c cos A + A + zx cac a cos B + + xy aba b cos C + 1 (x aa + y bb + z cc. (4. From cos A A 1, cos B 1, cos C 1, and x + y + z xy + yz + zx, we obtain yz bcb c cos A A + zx cac a cos B + xy aba b cos C yz bcb c + zx cac a + xy aba b x aa + y bb + z cc. (4. Inequality (1. follows from inequalities (4.1-(4.3 immediately. Thus, we complete the proof of Conjecture.
6 198 Yu-Lin Wu REFERENCES [1] An, Z.-P., Discussing about an Important Embedding Inequality in Triangle, Maths Teaching in Middle Schools, (5(1994, (in Chinese [] Bottema, O., R. Ž. Djordević, R. R. Janić, D. S. Mitrinović and P.M.Vasić, Geometric Inequality, Wolters-Noordhoff Publishing, Groningen, The Netherlands, [3] Pham Huu Duc, An Unexpectedly Useful Inequality, Mathematical Reflections, 3(1(008. [4] Gao, L., A New Inequality involved Two Triangles, Xiamen Mathematical Communications, (3(1983, 9. (in Chinese [5] Tran Quang Hung, On Some Geometric Inequalities, Mathematical Reflections, 3(3(008. [6] Jiang, W.-D., The Proof Of CIQ.131, Communications in Studies of Inequalities, 1(1(005, (in Chinese [7] Kuang, J.-C., Chángyòng Bùděngshì (Applied Inequalities, 3rd ed., Shandong Science and Technology Press, Jinan City, Shandong Province, China, 004, 9. (in Chinese [8] Leng, G.-S., Geometric Inequalities, East China Normal University Press, Shanghai City, China, 005, (in Chinese [9] Liu, J., Nine Sine Inequality, manuscript, 008, 77. (in Chinese [10] Liu, J., Inequalities involved Interior Bisectors of Angles in Triangle, Forward Position of Elementary Mathematics, 1996, (in Chinese [11] Pedoe, D., An Inequality for two triangles, Proc Cambridge Philos. Soc., 38(194, [1] Wu, Y.-S., The Proof of Whc143, High-School Mathematics Monthly, 0(9(1997, 40. (in Chinese [13] Yang, X.-Z., Research in Inequalities, Tibet People s Press, Lhasa, 000, 574. (in Chinese [14] Zhang, Z.-M., Pedoe s Inequality and Another Inequality involved Two Triangles, Bulletin of Maths, (1(1980, 8. (in Chinese [15] Octogon Mathematical Magazine ( Department of Mathematics, Beijing University of Technology 100 Pingleyuan, Chaoyang District, Beijing 10014, P.R.China ylwu198@gmail.com
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