Acute Triangle
An 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 $90^\circ$; equivalently, for sides $a, b, c$ with $c$ the largest, the triangle is acute if and only if $a^2+b^2>c^2$ for all three side combinations (all three dot products of adjacent side vectors are positive). An equilateral triangle is always acute, and its circumcenter, incenter, centroid, and orthocenter all lie inside the triangle; the orthocenter $H$, centroid $G$, and circumcenter $O$ are collinear on the Euler line, with $OG:GH = 1:2$.
Example
A triangle with angles $60^\circ$, $70^\circ$, and $50^\circ$ is acute since all are less than $90^\circ$; if even one angle reaches $90^\circ$ or more, it is not acute. For sides $5, 6, 7$: the largest side is $7$, and $5^2+6^2=61>49=7^2$, with the other two checks ($25+49>36$, $36+49>25$) also passing, confirming all angles are acute. The nine-point circle (radius $R/2$) of such a triangle passes through the midpoints of the sides, the feet of the altitudes, and the midpoints of the segments from each vertex to the orthocenter.
Key Insight
An easy check: if all three angles look sharp, less than a right-angle corner, the triangle is acute, and the circumcenter of an acute triangle always lies inside it. The condition $a^2+b^2>c^2$ is the strict Pythagorean inequality, showing that a right triangle ($a^2+b^2=c^2$) is the exact boundary between acute and obtuse triangles. The Euler line theorem, that $O$, $G$, $H$ are collinear with $OG:GH=1:2$, is one of the most elegant results in triangle geometry, and acute triangles have a richer set of interior special points than obtuse triangles, whose circumcenter exits the triangle.