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Geometry ProblemLevel: Mathematics Education, High School, Honors Geometry, College.Click the figure below to see the complete problem 829.
Denote BF, BG by p & q resply. Use Stewart's Theorem to obtain: 125+x^2=6(p^2+5), 50+4x^2=6(q^2+8)and p/(p+5/p) =q/(q+8/q).Solve simultaneously to get x=5√(2.5) --- same as Atharva
Since DE//AC so Arc(AD)=Arc(EC) => angle (ABF)=angle(CBG)So in triangle ABC ,line BF and BG are isogonal to each other.Per Steiner’s theorem for isogonal lines BF and BG (AF/FC).(AG/GC)=AB^2/BC^2So 1/5 . 4/2 =25/x^2So x=5.sqrt(5/2)