Proof without words of a simple conjecture about any triangle

Observe a spiral similarity with center at $D$ which takes $A$ to $K$. Then it takes $B$ to $M$ and $C$ to $L$. So this map takes triangle $ABC$ to triangle $KML$ which means that they are similary with dilatation coefficient $k={\sqrt{2}\over 2}$ So the ratio of the areas is $k^2 =1/2$.

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Geometry

Euclidean Geometry

Triangles

Geometric Construction

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