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Some extensions of a geometric inequality generated from Ptolemy’s inequality

Published online by Cambridge University Press:  15 October 2025

Quang Hung Tran
Quang Hung Tran*
Affiliation:
High School for Gifted Students, Hanoi University of Science, Vietnam National University, Hanoi, Vietnam e-mail: tranquanghung@hus.edu.vn
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Extract

For convenience, throughout this Article, we shall use the standard notations for any triangle ABC, where:

  1. 1) the lengths of the sides BC, CA, AB are denoted by a, b, c, respectively,

  2. 2) the area of the triangle is denoted by S,

  3. 3) the semiperimeter of the triangle is denoted by s,

  4. 4) the circumradius and inradius of the triangle are denoted by R and r, respectively,

  5. 5) the lengths of the medians corresponding to vertices A, B, C are denoted by ma, mb, mc, respectively,

  6. 6) the lengths of the angle bisectors corresponding to vertices A, B, C are denoted by la, lb, lc, respectively,

  7. 7) the lengths of the altitudes corresponding to vertices A, B, C are denoted by ha, hb, hc, respectively.

Information

Type
Articles
Information
The Mathematical Gazette , Volume 109 , Issue 576 , November 2025 , pp. 507 - 512
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Copyright
© The Authors, 2025 Published by Cambridge University Press on behalf of The Mathematical Association

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References

Tho, N. X., 106.27 An interesting application of Ptolemy’s inequality, Math. Gaz., 106 (July 2022) pp. 338–340.10.1017/mag.2022.81CrossRefGoogle Scholar
Problem 11945, Amer. Math. Monthly., 123(10) (December 2006) p. 1050.Google Scholar
Problem 102.E, Math. Gaz., 102 (July 2018) p. 355.CrossRefGoogle Scholar
Lukarevski, M., An alternate proof of Gerretsen’s inequalities, Elem. Math., 72(1) 2017, pp. 2–8.10.4171/em/317CrossRefGoogle Scholar