Obtuse Triangle Divided into Acute Triangles

Theorem
Let $T$ be an obtuse triangle.

Let $T$ be dissected into $n$ of acute triangles.

Then $n \ge 7$.

Construction
Let $\triangle ABC$ be an obtuse triangle such that:
 * $\angle C > 90^\circ$
 * $\angle C - \angle A < 90^\circ$
 * $\angle C - \angle B < 90^\circ$

Let $D$ be the incenter of $\triangle ABC$.

Let a circle be constructed whose center is $D$ and whose radius is $CD$.

The dissection into acute triangles is then performed as follows:


 * ObtuseTriangleAcuteDissection.png

This gives a dissection into $7$ pieces.

If the conditions on $A$ and $B$ are not satisfied, then it is possible to draw a line from $C$ to $AB$ to dissect $\triangle ABC$ into an acute triangle and an obtuse triangle which does satisfy the conditions.

This gives a dissection into $8$ pieces.