A conjecture about the sum of the areas of three triangles built on the sides of any given triangle

Draw the orthocenter. You get three parallelograms which immediately provide the answer.


The answer of Moti is perfect. Note, though, that this also means that this property holds only if the orthocenter is inside the triangle, otherwise the external triangles overlap and the required property does not hold any more.

Internal orthocenter.

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External orthocenter.

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Tags:

Geometry

Euclidean Geometry

Triangles

Geometric Construction

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