MHB Why do similar triangles have equal ratios of sides?

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Similar triangles maintain equal ratios of corresponding sides due to their equiangular nature, meaning their angles are congruent. This property leads to proportional relationships among the sides, which can be demonstrated through geometric proofs. The discussion also connects this concept to trigonometric functions, noting that the sine and cosine values for a given angle remain constant, reflecting the fixed ratios in similar triangles. The slope of a line, represented by the tangent of the angle it forms with the x-axis, further illustrates these relationships. Understanding these connections enhances comprehension of both triangle similarity and trigonometric principles.
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I was wondering why similar triangles have the property that the ratios of the similar sides are equal.
Or why the triangular functions (sin, cos,...) for a certain angle is fixed.
They are related, and if I can find one of them, the other can be proved easily.
I was thinking about the slope of the straight line since it is fixed and it is equal to the tan(angle which the line made with the positive x-axis) .

Any ideas?

Thanks.
 
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... You can just prove buy saying that angle AB,for example, is equal to AB2
 
Amer said:
I was wondering why similar triangles have the property that the ratios of the similar sides are equal.

Hi Amer, :)

Refer either of the following links.

1) Equiangular Triangles are Similar - ProofWiki

2) http://farside.ph.utexas.edu/euclid/Elements.pdf (Page 160)
 

Thread 'Area of Overlapping Squares'

Here is a little puzzle from the book 100 Geometric Games by Pierre Berloquin. The side of a small square is one meter long and the side of a larger square one and a half meters long. One vertex of the large square is at the center of the small square. The side of the large square cuts two sides of the small square into one- third parts and two-thirds parts. What is the area where the squares overlap?
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