Euclid Book 6 Proposition 19
Statement
If two triangles (△ABC, △DEF) are similar, then the ratio of their areas is equal to the ratio of the squares of their corresponding sides ().
the area of △ABC
the area of △DEF
2
AB
2
DE
Computational Explanation
GeometricScene{{A.,B.,C.,D.,E.,F.},{}},{GeometricAssertion[{Triangle[{A.,B.,C.}],Triangle[{D.,E.,F.}]},"Similar"]},
Area[Triangle[{A.,B.,C.}]]
Area[Triangle[{D.,E.,F.}]]
2
EuclideanDistance[A.,B.]
EuclideanDistance[D.,E.]
Area[Triangle[{A.,B.,C.}]] Area[Triangle[{D.,E.,F.}]] 2 EuclideanDistance[A.,B.] EuclideanDistance[D.,E.] |
 |
Explanations
Let , be similar triangles having the angle at equal to the angle at , and such that, as is to , so is to , so that corresponds to ;
I say that the triangle has to the triangle a ratio duplicate of that which has to .
For let a third proportional be taken to , , so that, as is to , so is to ;and let be joined.
Since then, as is to , so is to , therefore, alternately, as is to , so is to .
But, as is to , so is to ; therefore also, as is to , so is to .
Therefore in the triangles, the sides about the equal angles are reciprocally proportional.
But those triangles which have one angle equal to one angle, and in which the sides about the equal angles are reciprocally proportional, are equal; therefore the triangle is equal to the triangle .
Now since, as is to , so is to , and, if three straight lines be proportional, the first has to the third a ratio duplicate of that which it has to the second, therefore has to a ratio duplicate of that which has to .
But, as is to , so is the triangle to the triangle ;therefore the triangle also has to the triangle a ratio duplicate of that which has to .
But the triangle is equal to the triangle ; therefore the triangle also has to the triangle a ratio duplicate of that which has to .
ABC
DEF
B
E
AB
BC
DE
EF
BC
EF
I say that the triangle
ABC
DEF
BC
EF
For let a third proportional
BG
BC
EF
BC
EF
EF
BG
AG
Since then, as
AB
BC
DE
EF
AB
DE
BC
EF
But, as
BC
EF
EF
BG
AB
DE
EF
BG
Therefore in the triangles
ABG
DEF
But those triangles which have one angle equal to one angle, and in which the sides about the equal angles are reciprocally proportional, are equal; therefore the triangle
ABG
DEF
Now since, as
BC
EF
EF
BG
BC
BG
CB
EF
But, as
CB
BG
ABC
ABG
ABC
ABG
BC
EF
But the triangle
ABG
DEF
ABC
DEF
BC
EF