Let $\Delta ABC$ be a triangle in the plane. Let $P\_{1}, P\_{2}, P\_{3}$ be the intersection points of bisectors, medians and altitudes, respectively. We define the quantity:
\begin{equation}
Q(\Delta ABC)=\frac{\mathcal{A}(\Delta P{1}P\_{2}P\_{3})}{\mathcal{A}(\Delta ABC)}
\end{equation}  

where $\mathcal{A}$ is the area of a triangle.



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> Is it true to say that $\;$ $\sup \{Q(\Delta ABC)\mid \; \Delta ABC\;\text{varies among all triangles}\}<1$?(strictly)
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> If yes, what type of triangles assumes this supremum?
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> 



A quantity associated to a triangle
I can say something for the triangles with acute angles. But the obtuse angles may be a bit of a problem, since the intersection of the altitudes will be outside of the triangle.


For the triangle with acute angles, the supremum is definitely less than $1$. In fact, it would be true for any $P\_1$ and $P\_3$, so long as $P\_2$ is the intersection of the medians. Indeed, if you drop the condition that $P\_1$ and $P\_3$ are intersection of bisectors and altitudes, then the supremum is independent of the triangle. More specifically, since area is linear in $P\_1$ the supremum would be achieved when $P\_1$ is a vertex of $ABC$, and similar for $P\_3$. Thus, the ratio of the areas is less than $\frac 13$. 


