One can argue this question can in geometry, but this property of the Arbelo is elementary and a good foundation for intuitive and observational proofs, so show that that the length of the lower boundary of the arbelos equals the length upper boundary?

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One can argue this question can in geometry, but this property of the Arbelo is elementary and a good foundation for intuitive and observational proofs, so show that that the length of the lower boundary of the arbelos equals the length upper boundary?

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Answer 1 · 2016-10-24T16:54:05

Calling $hat(AB)$ the semicircumference length with radius $r$ , $hat(AC)$ the semicircumference length of radius $r_1$ and $hat(CB)$ the semicircumference length with radius $r_2$

We know that

$hat(AB) = lambda r$ , $hat(AC) = lambda r_1$ and $hat(CB)= lambda r_2$ then

$hat(AB)/r=hat(AC)/r_1=hat(CB)/r_2$ but

$hat(AB)/r = (hat(AC)+hat(CB))/(r_1+r_2) = (hat(AC)+hat(CB))/r$

because if

$n_1/n_2=m_1/m_2 = lambda$ then

$lambda =( n_1pmm_1)/(n_2pmm_2) = (lambda n_2pm lambda m_2)/(n_2pmm_2) = lambda$

so

$hat(AB)= hat(AC)+hat(CB)$

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One can argue this question can in geometry, but this property of the Arbelo is elementary and a good foundation for intuitive and observational proofs, so show that that the length of the lower boundary of the arbelos equals the length upper boundary?

1 Answer

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