People understood the concept of a curve intuitively in ancient times. By observing the trajectory of a thrown stone, the curvature of riverbanks, the contours of leaves of plants and flowers, people gradually developed the concept of curve. The paper attempts to discuss such curves, which have led mathematicians to take a deeper look at the concept of curve and to study the properties of curves more seriously. The authors use the term ``remarkable curves'' for those curves of third and fourth order that give rise to circles (generating circles). The equations of such curves in Cartesian coordinates can be derived using elementary geometry methods, usually using the similarity of triangles. The authors believe that the ideas presented in this article will help mathematics teachers to encourage students to take a more serious interest in the very interesting science of mathematics.
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