A family of quartics associated with a triangle - Peter Yff It is known that the envelope of the family of pedal lines (Simson or Wallace lines) of a triangle ABC is Steiner's deltoid, a three-cusped hypocycloid that is concentric with the nine-point circle of ABC and touches it at three points. Also known is that the nine-point circle is the locus of the intersection point of two perpendicular pedal lines. This paper considers a generalization in which two pedal lines form any acute angle \theta. It is found that the locus of their intersection point, for any value of \theta, is a quartic curve with the same axes of symmetry as the deltoid. Moreover, the deltoid is the envelope of the family of quartics. Finally, it is shown that all of these quartics, as well as the deltoid and the nine-point circle, may be simultaneously generated by points on a circular disk rolling on the inside of a fixed circle. ✪ ✪ ✪ ✪ ✪ Link tải tài liệu: LINK TẢI TÀI LIỆU Theo LTTK Education
