Online Geometry theorems, problems, solutions, and related topics.
Geometry Problem. Post your solution in the comments box below.Level: Mathematics Education, High School, Honors Geometry, College.Click the diagram below to enlarge it.
It is equivalent to show that the srea of the three squares is equal to the blue rectangle. Algebraically, to show that EA² + EA×EB + EB² = 3×OD²Now considerOD² = OA²= OO₁² + [(EA + EB)/2]²= (EO₁ tan30°)² + [(EA + EB)/2]²= 1/3×[(EA − EB)/2]² + [(EA + EB)/2]²= 1/3×[EA² + EA×EB + EB²]Hence, EA² + EA×EB + EB² = 3×OD².
Let AD=AE= l , EB=d and OD= RWe need to that l^2+d^2+l.d=3.R^2In triangle ABD we have BD/sin(60)= 2.R => BD= R.SQRT(3)Apply cosine formula in triangle ABD we haveBD^2=AD^2+AB^2-2.AB.AD.cos(60) => l^2+(l+d)^2-l(l+d)So 3.R^2=l^2+d^2+l.d or Blue area= yellow areas