Prove Similar Triangles: $\sqrt{aa_1}+\sqrt{bb_1}+\sqrt{cc_1}$

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In summary, the formula $\sqrt{aa_1}+\sqrt{bb_1}+\sqrt{cc_1}$ represents the sum of the square roots of corresponding sides of two similar triangles. It can be used to prove two triangles are similar by showing that their corresponding sides have equal sums of square roots. The use of square roots is significant because they represent side lengths in right triangles. However, this formula can only be used to prove similarity if the corresponding sides are proportional. This formula is related to the Pythagorean theorem because it uses the concept of right triangles and their side lengths to prove similarity.
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Prove that two triangles with sides $a,\,b,\,c$ and $a_1,\,b_1,\,c_1$ are similar if and only if $\sqrt{aa_1}+\sqrt{bb_1}+\sqrt{cc_1}=\sqrt{(a+b+c)(a_1+b_1+c_1)}$.
 
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anemone said:
Prove that two triangles with sides $a,\,b,\,c$ and $a_1,\,b_1,\,c_1$ are similar if and only if $\sqrt{aa_1}+\sqrt{bb_1}+\sqrt{cc_1}=\sqrt{(a+b+c)(a_1+b_1+c_1)}$.
we have
$\sqrt{aa_1}+\sqrt{bb_1}+\sqrt{cc_1}=\sqrt{(a+b+c)(a_1+b_1+c_1)}$.

$\equiv (\sqrt{aa_1}+\sqrt{bb_1}+\sqrt{cc_1})^2=(a+b+c)(a_1+b_1+c_1)$

$\equiv aa_1+bb_1+cc_1+2\sqrt{aa_1bb_1} + 2\sqrt{bb_1cc_1} + 2\sqrt{cc_1aa_1} = aa_1+ab_1 + ac_1 + ba_1 + bb_1 + bc_1 + ca_1 + cb_1 + cc_1$

$\equiv 2\sqrt{aa_1bb_1} + 2\sqrt{bb_1cc_1} + 2\sqrt{cc_1aa_1} = ab_1 + ac_1 + ba_1 + bc_1 + ca_1 + cb_1$

$\equiv ab_1 + ac_1 + ba_1 + bc_1 + ca_1 + cb_1-2(\sqrt{aa_1bb_1} + 2\sqrt{bb_1cc_1} + 2\sqrt{cc_1aa_1}) = 0$

$\equiv (\sqrt{ab_1} - \sqrt{a_1b})^2 + (\sqrt{ac_1} - \sqrt{a_1c})^2 + (\sqrt{bc_1} - \sqrt{b_1c})^2 = 0$

The above is true iff $ab_1 = a_1b$, $ac_1 = a_1c$, $bc_1 = b_1c$

giving $\frac{a}{a_1} = \frac{b}{b_1} = \frac{c}{c_1}$ or the 2 triangles are similar
 
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Related to Prove Similar Triangles: $\sqrt{aa_1}+\sqrt{bb_1}+\sqrt{cc_1}$

What is the formula for proving similar triangles using $\sqrt{aa_1}+\sqrt{bb_1}+\sqrt{cc_1}$?

The formula for proving similar triangles using $\sqrt{aa_1}+\sqrt{bb_1}+\sqrt{cc_1}$ is:$$\frac{a}{a_1}=\frac{b}{b_1}=\frac{c}{c_1}$$

How does the formula for proving similar triangles using $\sqrt{aa_1}+\sqrt{bb_1}+\sqrt{cc_1}$ work?

The formula works by comparing the corresponding sides of two triangles. If the ratios of the corresponding sides are equal, then the triangles are similar.

What is the significance of using square roots in the formula for proving similar triangles?

The use of square roots in the formula helps to account for the lengths of the sides of the triangles, rather than just the ratios. This allows for a more accurate determination of similarity.

Can the formula for proving similar triangles using $\sqrt{aa_1}+\sqrt{bb_1}+\sqrt{cc_1}$ be used for all types of triangles?

Yes, the formula can be used for all types of triangles, including right triangles, acute triangles, and obtuse triangles.

Are there any limitations to using the formula for proving similar triangles using $\sqrt{aa_1}+\sqrt{bb_1}+\sqrt{cc_1}$?

One limitation is that the formula only works for triangles, and cannot be applied to other types of polygons. Additionally, the formula may not be as useful for triangles with very small or large side lengths, as the square roots may become difficult to calculate accurately.

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