Acute and obtuse triangles
triangle without a right angle
An acute triangle (or acute-angled triangle) is a triangle with three acute angles (less than 90°). An obtuse triangle (or obtuse-angled triangle) is a triangle with one obtuse angle (greater than 90°) and two acute angles. Since a triangle's angles must sum to 180° in Euclidean geometry, no Euclidean triangle can have more than one obtuse angle.
Acute and obtuse triangles are the two different types of oblique triangles—triangles that are not right triangles because they do not have any right angles (90°).
Properties
In all triangles, the centroid—the intersection of the medians, each of which connects a vertex with the midpoint of the opposite side—and the incenter—the center of the circle that is internally tangent to all three sides—are in the interior of the triangle. However, while the orthocenter and the circumcenter are in an acute triangle's interior, they are exterior to an obtuse triangle.
The orthocenter is the intersection point of the triangle's three altitudes, each of which perpendicularly connects a side to the opposite vertex. In the case of an acute triangle, all three of these segments lie entirely in the triangle's interior, and so they intersect in the interior. But for an obtuse triangle, the altitudes from the two acute angles intersect only the extensions of the opposite sides. These altitudes
fall entirely outside the triangle, resulting in their intersection with each other (and hence with the extended altitude from the obtuse-angled vertex) occurring in the triangle's exterior.
“Acute and obtuse triangles” enters the record as triangle without a right angle. Crown Archives preserves that source wording while asking what Acute, obtuse and triangles can confirm, complicate or overturn.
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This entry incorporates text from “Acute and obtuse triangles” on English Wikipedia. Contributors are listed in the page history. Text is available under the Creative Commons Attribution-ShareAlike 4.0 License. Selected authority identifiers and statements are retrieved from Wikidata under CC0; their references and qualifiers remain part of the verification path.