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Post by mondo on Oct 23, 2022 21:42:56 GMT
I wonder how do we know that in figure [2c] page 126 and especially in statement (5) we deduce that in the inversion side $q\tilde{b}$ corresponds to $qa$? In the book author seems to rely on marked angles on the figure but this is not precise. So how are w sure? It is a bit counterintuitive, at first I thought, $\tilde{a}q$ corresponds to $aq$, likewise $\tilde{b}q$ corresponds to $bq$ due to the proportion between them.
Thank you.
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Post by Admin on Oct 23, 2022 22:40:04 GMT
I wonder how do we know that in figure [2c] page 126 and especially in statement (5) we deduce that in the inversion side $q\tilde{b}$ corresponds to $qa$? In the book author seems to rely on marked angles on the figure but this is not precise. So how are w sure? It is a bit counterintuitive, at first I thought, $\tilde{a}q$ corresponds to $aq$, likewise $\tilde{b}q$ corresponds to $bq$ due to the proportion between them. Thank you. Mondo The whole of the proof is in the book and is based on the rules of inversion. All you need to know are the 3 rules for defining similar triangles to see that the triangles must be similar. Vasco |
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Post by mondo on Oct 23, 2022 23:01:36 GMT
Ok I figured that out. Thanks!
For the record, just because pairs of points were inverted in the same way doesn't mean they are in proportion to each other as I originally assumed. Thankfully, we have a relation for $R^2$ from which we can deduce which sides are in proportion and hence figure out the similarity of the triangles and especially, corresponding sides/angles.
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