Find three ways to divide any acute triangle into $3$ mirror-symmetric shapes.

Two ways, using the circumcenter and incenter, are easy.


What is the third way?

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    $\begingroup$ I strongly suspect the three resulting portions are not all of the same generic type, e.g., isosceles triangles. $\endgroup$ – David G. Stork Apr 21 '19 at 3:12

Take triangle $ABC$, let $AH$ be the altitude. Let $M$ be the midpoint of $AB$, $N$ the midpoint of $AC$. Then $ABC$ can be decomposed into isosceles triangle $BMH$, isosceles triangle $HNC$, and the quadrilateral $AMHN$ which has symmetric axis $MN$.

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    $\begingroup$ Yep. Nice. Confirming my intuitions in my comment. (+1) $\endgroup$ – David G. Stork Apr 21 '19 at 3:31

In your second graph for incenter, the 3 curved polygons $DIGH$, $EGI$ and $FHG$ are also mirror-symmetric and you can use them to decompose your triangle.

A triangle into 3 curved polygon


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