Acute Triangle
GeometryAn acute triangle has all three interior angles measuring less than 90 degrees.
Formula
a^2 + b^2 > c^2 \text{ (for all three side combinations)}
Definition
An acute triangle has all three angles smaller than $90^\circ$. Every single corner is an acute (sharp) angle.
Example
A triangle with angles $60^\circ$, $70^\circ$, and $50^\circ$ is acute (all less than $90^\circ$). An equilateral triangle is always acute because all angles are $60^\circ$. If even one angle is $90^\circ$ or more, it is NOT acute.
Key Insight
An easy check: if all three angles are sharp-looking (less than a right angle corner), the triangle is acute. The circumcenter of an acute triangle always lies inside the triangle.
Definition
An acute triangle has all three interior angles measuring less than $90^\circ$. Equivalently, for sides $a, b, c$ (largest side $c$): the triangle is acute iff $a^2 + b^2 > c^2$. The circumcenter lies inside an acute triangle. An equilateral triangle is always acute.
Example
Triangle with sides $5, 6, 7$: largest side $7$. Check: $5^2 + 6^2 = 25 + 36 = 61 > 49 = 7^2$. All three checks pass ($25+49>36$, $36+49>25$), confirming all angles are acute.
Key Insight
The condition $a^2 + b^2 > c^2$ for an acute triangle is the strict Pythagorean inequality. It shows that the right triangle ($a^2 + b^2 = c^2$) is the exact boundary between acute and obtuse triangles in the space of triangles.
Definition
An acute triangle satisfies $a^2 + b^2 > c^2$ for all three pairs of sides (equivalently, all three dot products of adjacent side vectors are positive). Its circumcenter, incenter, centroid, and orthocenter all lie in the triangle's interior. The orthocenter $H$, centroid $G$, and circumcenter $O$ are collinear (Euler line): $OG:GH = 1:2$.
Example
For the acute triangle with sides $5, 6, 7$: the Euler line passes through the circumcenter, centroid, and orthocenter in ratio $1:2$. The nine-point circle (radius $R/2$) passes through the midpoints of sides, feet of altitudes, and midpoints of vertex-to-orthocenter segments.
Key Insight
Acute triangles have a richer set of interior special points than obtuse triangles (circumcenter exits the triangle for obtuse). The Euler line theorem, stating $O, G, H$ are collinear with $OG:GH = 1:2$, is one of the most elegant results in triangle geometry and generalizes to the nine-point center as the midpoint of $OH$.