# Divide an acute triangle into $3$ mirror-symmetric shapes

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?

• I strongly suspect the three resulting portions are not all of the same generic type, e.g., isosceles triangles. – 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 $$MN$$.
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.