I have seen multiple sites where it is proved that the ratio of corresponding sides of two similar triangles is constant. But the thing is they have used trigonometry in that case. As far as I know, trigonometry is itself based on the fact that the ratio corresponding sides of similar triangles is constant. So, I am interested in knowing about some algebraic way to prove it.
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3What's your definition of smilar triangles? – Nov 30 '19 at 13:31
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triangles whose corresponding angles are equal. – vighnesh153 Nov 30 '19 at 13:32
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3Make a vertex and the two sides at that vertex overlap. The third sides will be parallel. The proportionality is Thales' theorem. See, here a proof of Thales' theorem. – conditionalMethod Nov 30 '19 at 13:38
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This is somehow equivalent to the Thales theorem isn't? – Jeanba Nov 30 '19 at 13:48
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The standard definition is: two triangles are similar if the lengths of corresponding sides are proportional. This is actually equivalent to the assertion that corresponding angles are equal, as it was proved (without trigonometry) in Euclid's Elements (Book VI, Propositions 4–5).
https://mathcs.clarku.edu/~djoyce/java/elements/bookVI/propVI4.html
https://mathcs.clarku.edu/~djoyce/java/elements/bookVI/propVI5.html
José Carlos Santos
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