5 problems

Let $X_k'$ denote a triangle center of the polar family, and suppose its locus is a circle with nonzero radius. Circular-locus pencil conjecture. That circle belongs to the parabol…

Let $P_1P_2P_3$ be a bic-III family of triangles inscribed in an outer circle $\mathcal{C}$, with each side tangent to a distinct in-pencil circle, and assume all four in-pencil ci…

Let $P_1P_2P_3$ be a bic-III family of triangles inscribed in an outer circle $\mathcal{C}$, with each side tangent to a distinct in-pencil circle, and assume all four in-pencil ci…

Let $P_1P_2P_3$ be a bic-III family of triangles inscribed in an outer circle $\mathcal{C}$, with $P_1P_2$ and $P_1P_3$ tangent to distinct circles $\mathcal{C}'$ and…

Let $X$ be a triangle center that does not always lie on a triangle's circumcircle. Consider the bic-II family and its bic-I (poristic) subfamily. A locus is stationary when the co…