Dividing a triangle into four congruent triangles. Proof how this works?

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So for any triangle, you can divide it into four congruent triangles by connecting the midpoints of each side. But I want to see how this works.

How does SR can be proved to be equal to AQ? SQ equal to RC? RQ equal to AS?

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1 Answer

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$S$ is the midpoint of $AB$, so $|AS| = |SB| = \frac{|AB|}{2}$. Similarly for $R$, $|BR| = |RC| = \frac{|BC|}{2}$. Also, $\angle ABC = \angle SBR$, so by SSA similarity $\triangle BSR \sim \triangle ABC$. The common ratio is $1:2$, so $|SR| = \frac{|AC|}{2} = |AQ| = |QC|$. The same goes for $\triangle CQR$ and $\triangle ASQ$.

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