

Problem 524: Circle, Equilateral Triangles, Midpoint, Side, Measurement. Level: High School, SAT Prep, College geometry

The figure shows an equilateral
triangle ABC (side a) inscribed in a circle C_{1}. The point D is the midpoint
of BC and E and F are on the circle C_{1}. If the
triangle DEF is equilateral (side x), prove that
.
Post a comment or solution.


References:


Reference: Fukagawa Hidetoshi, Tony Rothman, "Sacred Mathematics: Japanese Temple Geometry" (Princenton
University Press, 2008).
Between the seventeenth and nineteenth centuries Japan was totally isolated from the West by imperial decree. During that time, a unique brand of homegrown mathematics flourished, one that was completely uninfluenced by developments in Western mathematics. People from all walks of life—samurai, farmers, and merchants—inscribed a wide variety of geometry problems on wooden tablets called sangaku and hung them in Buddhist temples and Shinto shrines throughout Japan. 

Problem 525.
Circles, Diameter, Tangent, Radius, Congruence, Measurement. 

Problem 523.
Tangent Circles, Diameter Perpendicular, Collinearity. 

Problem 522.
Right Triangle, Circle, Diameter, Tangent. 








