= Similar triangle
{title2=$\triangle ABC\sim\triangle A'B'C'$}
Two <triangles> are similar if a <Euclidean similarity> maps one onto the other. Their corresponding <angles> agree, and corresponding side <lengths> have a common positive ratio.
Specify the correspondence first: $A$ matches $A'$, $B$ matches $B'$ and $C$ matches $C'$. For <similar triangles>, the common scale factor $k>0$ satisfies
$$
\frac{|A'B'|}{|AB|}=\frac{|B'C'|}{|BC|}=\frac{|C'A'|}{|CA|}=k.
$$
The <ratios> compare matching sides in the same order. A <Euclidean similarity> can also move, turn or reflect the <triangle>; its orientation on the page is irrelevant. In the <Euclidean plane>, matching just two interior <angles> is enough: their sums force the third to match, and equal corresponding <angles> force the side <ratios> to agree. This is the angle-angle (AA) criterion for <similar triangles>. Matching just one <angle> is insufficient.
Scale the <right triangle> with vertices $(0,0)$, $(4,0)$ and $(0,3)$ by $k=2$. Its new vertices are $(0,0)$, $(8,0)$ and $(0,6)$, and its side <lengths> are $6,8,10$. The matching side <ratios> are $6/3=8/4=10/5=2$. Both <triangles> have a <right angle> and the same two other interior <angles>, approximately $36.87^\circ$ and $53.13^\circ$. However, their <areas> are $3\cdot4/2=6$ and $6\cdot8/2=24$: doubling <lengths> multiplies <area> by four.
\Image[/similar-triangle-scaling.png]
{title=Similar 3-4-5 and 6-8-10 right triangles: side lengths multiply by $2$, while area multiplies by $2^2=4$}
Generally, a <Euclidean similarity> multiplies both a base and its perpendicular height by $k$, so the <area> becomes $\tfrac12(kb)(kh)=k^2\tfrac12bh$. This explains why an enlarged drawing needs four times as much surface when each <length> doubles. It also enables indirect measurement: under assumptions of level ground, vertical objects and parallel light rays, an object of height $3$ casting a shadow of <length> $4$ provides a <right triangle>. Another object's shadow of <length> $8$ gives a <similar triangle> and height $3(8/4)=6$. The shared <angles> justify the inference; visual resemblance alone does not.
<Euclid's Elements> VI, Definition 1 describes similarity using corresponding <angles> and proportional sides. VI.4 establishes the implication from matching <angles> to matching side <ratios>, while VI.19 states the squared relation between side <ratios> and <areas>. These are documented ancient formulations of the principles used above, without establishing a first discoverer.https://mathcs.clarku.edu/~djoyce/elements/bookVI/defVI1.html[Euclid, Elements VI, Definition 1]{ref}https://mathcs.clarku.edu/~djoyce/elements/bookVI/propVI4.html[Euclid, Elements VI.4]{ref}https://mathcs.clarku.edu/~djoyce/elements/bookVI/propVI19.html[Euclid, Elements VI.19]{ref}
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