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.
2026-04-03 04:23:45.1775190225
why do similar triangles have proportional sides?
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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