Euclid Book 6 Proposition 15a
Statement
Two equal area triangles (△ACB, △DFE) with a pair of equal angles (∠ACB, ∠DFE), have the sides adjacent to the equal angles inversely proportional ().
CA
FD
FE
CB
Computational Explanation
GeometricScene{{A.,B.,C.,D.,E.,F.},{}},{Area[Triangle[{A.,C.,B.}]]Area[Triangle[{D.,F.,E.}]],PlanarAngle[{A.,C.,B.}]PlanarAngle[{D.,F.,E.}]},
EuclideanDistance[C.,A.]
EuclideanDistance[F.,D.]
EuclideanDistance[F.,E.]
EuclideanDistance[C.,B.]
EuclideanDistance[C.,A.] EuclideanDistance[F.,D.] EuclideanDistance[F.,E.] EuclideanDistance[C.,B.] |
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Explanations
Let , be equal triangles having one angle equal to one angle, namely the angle to the angle ; I say that in the triangles , the sides about the equal angles are reciprocally proportional, that is to say, that, as is to , so is to .
For let them be placed so that is in a straight line with ; therefore is also in a straight line with .
Let be joined.
Since then the triangle is equal to the triangle , and is another area, therefore, as the triangle is to the triangle , so is the triangle to the triangle .
But, as is to , so is to ,and, as is to , so is to .[id.]
Therefore also, as is to , so is to .
Therefore in the triangles, the sides about the equal angles are reciprocally proportional.
ABC
ADE
BAC
DAE
ABC
ADE
CA
AD
EA
AB
For let them be placed so that
CA
AD
EA
AB
Let
BD
Since then the triangle
ABC
ADE
BAD
CAB
BAD
EAD
BAD
But, as
CAB
BAD
CA
AD
EAD
BAD
EA
AB
Therefore also, as
CA
AD
EA
AB
Therefore in the triangles
ABC
ADE