During the IMC 2019 contest i ended up with the following question in elementary 2D Euclidean geometry: Let $\mathrm{CDE}$ be any nondegenerate triangle inside a circle, consider the regions with areas labelled $S_1, S_2, S_3$ as on the picture. Prove that: $$S_1 + S_2 + S_3 > S(CDE)$$ It must be known and i would be happy if anyone knows the elementary solution (it seems that i can d it by rough 5-pages calculations with cases).
It is also interesting to find the best possible constant $\lambda$ such that: $$S_1 + S_2 + S_3 \geq \lambda \cdot S(CDE) $$ My guess is $\lambda = \frac{4\pi}{3\sqrt{3}} - 1\approx 1.418399$, that comes from equilateral inscribed triangle.
My proof is valid only when the circum-circle of CDE lies completely inside the original circle.
Locate H, the ortho-center of CDE.
EH extended will cut CDE at E’ and the circum-circle at E’’
By properties of the ortho-center, HE’ = E’E’’. Hence, [CHE’] = [CE’E’’]
The other parts of CDE can be off-set similarly. Result follows.