Explore Japanese temple geometry problems in this 46-minute lecture from the "Famous Math Problems" series. Delve into a problem posed by 13-year-old Sato Naosue in 1847, examining it from both classical and Rational Trigonometry perspectives. Discover the surprising complexity of the algebra involved and gain insights into the symmetry between a triangle's incenter and excenters. Learn about the unique relationship between incenter quadrances in a 3-4-5 triangle, providing motivation to further study Rational Trigonometry as a modern approach to geometry. Follow along with the step-by-step solution, from the initial problem statement through hints, classical solutions, and advanced algebraic manipulations.