Problem
GEO-B3-M06-P022 Isogonal of a Kiepert Point
In triangle \(ABC\), similar triangles with parameter \(\varphi\) are built on the sides, and the corresponding cevians meet at \(X_\varphi\). Prove that the isogonal conjugate has trilinear coordinates proportional to \((\sin(A+\varphi):\sin(B+\varphi):\sin(C+\varphi))\).
C. Hint 1. Write equations of two cevians in trilinear coordinates.
D. Hint 2. Isogonal conjugation in trilinears sends \((x:y:z)\) to \((1/x:1/y:1/z)\).
For the cevian from \(C\) to the vertex of the constructed triangle on \(AB\), the construction angles give an equation of the form \(x\sin(A+\varphi)=y\sin(B+\varphi)\). The other two cevians give cyclic equations.
Solving them gives trilinear coordinates of \(X_\varphi\) reciprocal to \(\sin(A+\varphi),\sin(B+\varphi),\sin(C+\varphi)\), up to a common factor. Isogonal conjugation in trilinears replaces coordinates by reciprocals. Hence the isogonal point has coordinates \((\sin(A+\varphi):\sin(B+\varphi):\sin(C+\varphi))\).
This previews the Kiepert hyperbola; the full curve equation can be left for later.