Geometry Problem. Post your solution in the comments box below.

Level: Mathematics Education, High School, Honors Geometry, College.

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## Thursday, December 25, 2014

### Geometry Problem 1070: Circle, Chord, Equilateral Triangle, Square, Rectangle, Area, Diameter, Perpendicular

Labels:
area,
circle,
diameter,
equilateral,
perpendicular,
rectangle,
square,
triangle

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It is equivalent to show that the srea of the three squares is equal to the blue rectangle.

ReplyDeleteAlgebraically, to show that

EA² + EA×EB + EB² = 3×OD²

Now consider

OD² = 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= R

ReplyDeleteWe need to that l^2+d^2+l.d=3.R^2

In triangle ABD we have BD/sin(60)= 2.R => BD= R.SQRT(3)

Apply cosine formula in triangle ABD we have

BD^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