Arbelos

Exercise your brain. Archimedes wrote the "Book of Lemmas" more than 2200 years ago. Solve the proposition #6 (high school level) and lift up your geometry skills.

Continue reading at:

gogeometry.com/ArchBooLem06.htm

## Tuesday, December 16, 2008

### Archimedes' Book of Lemmas, Proposition #6

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arbelo,
Archimedes,
book of lemmas,
circle,
diameter

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Let AC = 2a, CB = 2b, DE = 2k .

ReplyDeleteDenote the midpoints of AC, CB, AB, DE

by X, Y, O, Z respectively.

(These are respectively the centres of the circles o AC, CB,AB, DE as diameters)

Clearly AO = OB = a + b,

AX = XC = a, CY = YB = b and so

XO = b - a, XY = a + b, DE / AB = k / (a + b).

XZ = a + k, YZ = b + k, OZ = a + b -k

by tangency considerations

Also note a /b = 2a / 2b = AC / CB = r.

Let angle ZXY = x

Apply Cosine Rule for x in triangles XYZ, XOZ

2(a + k)(a + b)cos x

= (a + k)^2 + (a + b)^2 - (b + k)^2

= 2k(a - b)+ 2a(a + b)

In the same manner

2(a + k)(b)cos x

= (a + k)^2 + b^2 - (a + b - k)^2

= 2k(2a + b)+ a(a - 2b)

Dividing respective sides;

(a + b )/ b

= [2k(a - b)+ 2a(a + b)]/[2k(2a + b)- 2ab]

implies

2bk(a - b)+ 2ab(a + b)

= 2k(a + b)(2a + b) - 2ab(a + b)

k(2ab - 2b^2 - 4a^2 - 6ab - 2b^2 ) = - 4ab(a + b)

k(- 4ab – 4a^2 – 4b^2) = -4ab(a + b)

k = ab(a + b ) / (a^2 + ab + b^2)

DE / AB = k / (a + b)

= ab / (a^2 + ab + b^2)

= 1 /[(a/b) + 1 + (b/a)]

= 1 / [r + 1 + (1/r)]

= r / (r^2 + r +1)