Shahrina Binti Ismail2024-05-282024-05-282018ISMAIL, S. (2020). PERFECT TRIANGLES ON THE CURVE $C_{4}$. Journal of the Australian Mathematical Society, 109(1), 68-80. doi:10.1017/S144678871900003X1446-78872243-210.1017/S144678871900003Xhttps://www.cambridge.org/core/journals/journal-of-the-australian-mathematical-society/article/abs/perfect-triangles-on-the-curve-c4/835262913F580360432210C2A1C957AEhttps://oarep.usim.edu.my/handle/123456789/7156Volume 109 Issue 1A Heron triangle is a triangle that has three rational sides (a,b,c) and a rational area, whereas a perfect triangle is a Heron triangle that has three rational medians (k,l,m) . Finding a perfect triangle was stated as an open problem by Richard Guy [Unsolved Problems in Number Theory (Springer, New York, 1981)]. Heron triangles with two rational medians are parametrized by the eight curves C1,…,C8 mentioned in Buchholz and Rathbun [‘An infinite set of heron triangles with two rational medians’, Amer. Math. Monthly 104(2) (1997), 106–115; ‘Heron triangles and elliptic curves’, Bull. Aust. Math.Soc. 58 (1998), 411–421] and Bácskái et al. [Symmetries of triangles with two rational medians, http://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.65.6533, 2003]. In this paper, we reveal results on the curve C4 which has the property of satisfying conditions such that six of seven parameters given by three sides, two medians and area are rational. Our aim is to perform an extensive search to prove the nonexistence of a perfect triangle arising from this curve.en-USelliptic curve, perfect triangle, p-adicArticle68801091