G.345 numbers continued proportion

IX.15

If three numbers in continued proportion be the least of those which have the same ratio with them, any two whatever added together is prime to the remaining number.

numbers_continued_proportion_ix_15(A, B, C)

given A, B, C, three numbers in continued proportion, be the least of those which have the same ratio with them

prove any two of the numbers A, B, C whatever added together are prime to the remaining number, namely A, B to C

given A, B, C, three numbers in continued proportion, be the least of those which have the same ratio with them

assert B, C to A

assert further A, C to B

construct find_numbers_continued: Let two numbers / D E /, / E F /, the least of those which have the same ratio with A, B, C, be taken.

∵ find_numbers_continuedassert It is then manifest that / D E / by multiplying itself has made A, and by multiplying / E F / has made B, and, further, / E F / by multiplying itself has made C

assert / D E /, / E F / are least,

∵ least_numbers_ratio_vii_22assert They are prime to one another

∵ numbers_prime_sumassert But, if two numbers be prime to one another, their sum is also prime to each

assert / D F / is also prime to each of the numbers / D E /, / E F /

assert Further / D E / is also prime to / E F /

assert / D F /, / D E / are prime to / E F /

∵ numbers_prime_number_vii_24assert But, if two numbers be prime to any number, their product is also prime to the other

∵ numbers_prime_productassert So that the product of / D F /, / D E / is prime to / E F / ; hence the product of / D F /, / D E / is also prime to the square on / E F /

∵ rectangle_whole_and_partassert The product of / D F /, / D E / is the square on / D E / together with the product of / D E /, / E F /

assert the square on / D E / together with the product of / D E /, / E F / is prime to the square on / E F /

assert The square on / D E / is A, the product of / D E /, / E F / is B, and the square on / E F / is C

assert A, B added together are prime to C

assert Similarly we can prove that B, C added together are prime to A

assert I say next that A, C added together are also prime to B

assert / D F / is prime to each of the numbers / D E /, / E F /,

∵ numbers_prime_number_vii_24, numbers_prime_productassert The square on / D F / is also prime to the product of / D E /, / E F /

∵ square_of_sumassert The squares on / D E /, / E F / together with twice the product of / D E /, / E F / are equal to the square on / D F /

assert the squares on / D E /, / E F / together with twice the product of / D E /, / E F / are prime to the product of / D E /, / E F /

assert :term:`Separando <G.197>`, the squares on / D E /, / E F / together with once the product of / D E /, / E F / are prime to the product of / D E /, / E F /

assert separando again, the squares on / D E /, / E F / are prime to the product of / D E /, / E F /

assert The square on / D E / is A, the product of / D E /, / E F / is B, and the square on / E F / is C

∴ A, C added together are prime to B

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