Synthetic and Vector Proofs for Two Hard Geometry Problems

Key Points

• To establish synthetic and vector proofs for two advanced problems in Euclidean geometry.
• Developed synthetic proofs for mixtilinear incircles and incenter-circumcenter properties.
• Constructed vector-based proofs to validate the findings.
• Demonstrated new insights into the properties of mixtilinear incircles.
• Confirmed the orthogonality between incenter and circumcenter using both proof methods.