Acute and obtuse triangles

If angle C is obtuse then for sides a, b, and c we have[4]^{: p.1, #74}

${\displaystyle {\frac {c^{2}}{2}}<a^{2}+b^{2}<c^{2},}$

with the left inequality approaching equality in the limit only as the apex angle of an isosceles triangle approaches 180°, and with the right inequality approaching equality only as the obtuse angle approaches 90°.

If the triangle is acute then

${\displaystyle a^{2}+b^{2}>c^{2},\quad b^{2}+c^{2}>a^{2},\quad c^{2}+a^{2}>b^{2}.}$

If C is the greatest angle and h_c is the altitude from vertex C, then for an acute triangle[4]^{: p.135, #3109}

${\displaystyle {\frac {1}{h_{c}^{2}}}<{\frac {1}{a^{2}}}+{\frac {1}{b^{2}}},}$

with the opposite inequality if C is obtuse.

With longest side c and medians m_a and m_b from the other sides,[4]^{: p.136, #3110}

${\displaystyle 4c^{2}+9a^{2}b^{2}>16m_{a}^{2}m_{b}^{2}}$

for an acute triangle but with the inequality reversed for an obtuse triangle.

The median m_c from the longest side is greater or less than the circumradius for an acute or obtuse triangle respectively:[4]^{: p.136, #3113}

${\displaystyle m_{c}>R}$

for acute triangles, with the opposite for obtuse triangles.

Ono's inequality for the area A,

${\displaystyle 27(b^{2}+c^{2}-a^{2})^{2}(c^{2}+a^{2}-b^{2})^{2}(a^{2}+b^{2}-c^{2})^{2}\leq (4A)^{6},}$

holds for all acute triangles but not for all obtuse triangles.

For an acute triangle we have, for angles A, B, and C,[4]^{: p.26, #954}

${\displaystyle \cos ^{2}A+\cos ^{2}B+\cos ^{2}C<1,}$

with the reverse inequality holding for an obtuse triangle.

For an acute triangle with circumradius R,[4]^{: p.141, #3167}

${\displaystyle a\cos ^{3}A+b\cos ^{3}B+c\cos ^{3}C\leq {\frac {abc}{4R^{2}}}}$

and[4]^{: p.155, #S25}

${\displaystyle \cos ^{3}A+\cos ^{3}B+\cos ^{3}C+\cos A\cos B\cos C\geq {\frac {1}{2}}.}$

For an acute triangle,[4]^{: p.115, #2874}

${\displaystyle \sin ^{2}A+\sin ^{2}B+\sin ^{2}C>2,}$

with the reverse inequality for an obtuse triangle.

For an acute triangle,[4]^{: p178, #241.1}

${\displaystyle \sin A\cdot \sin B+\sin B\cdot \sin C+\sin C\cdot \sin A\leq (\cos A+\cos B+\cos C)^{2}.}$

For any triangle the triple tangent identity states that the sum of the angles' tangents equals their product. Since an acute angle has a positive tangent value while an obtuse angle has a negative one, the expression for the product of the tangents shows that

${\displaystyle \tan A+\tan B+\tan C=\tan A\cdot \tan B\cdot \tan C>0}$

for acute triangles, while the opposite direction of inequality holds for obtuse triangles.

We have[4]^{: p.26, #958}

${\displaystyle \tan A+\tan B+\tan C\geq 2(\sin 2A+\sin 2B+\sin 2C)}$

for acute triangles, and the reverse for obtuse triangles.

For all acute triangles,[4]^{: p.40, #1210}

${\displaystyle (\tan A+\tan B+\tan C)^{2}\geq (\sec A+1)^{2}+(\sec B+1)^{2}+(\sec C+1)^{2}.}$

For all acute triangles with inradius r and circumradius R,[4]^{: p.53, #1424}

${\displaystyle a\tan A+b\tan B+c\tan C\geq 10R-2r.}$

For an acute triangle with area K, [4]^{: p.103, #2662}

${\displaystyle ({\sqrt {\cot A}}+{\sqrt {\cot B}}+{\sqrt {\cot C}})^{2}\leq {\frac {K}{r^{2}}}.}$

In an acute triangle, the sum of the circumradius R and the inradius r is less than half the sum of the shortest sides a and b:[4]^{: p.105, #2690}

${\displaystyle R+r<{\frac {a+b}{2}},}$

while the reverse inequality holds for an obtuse triangle.

For an acute triangle with medians m_a , m_b , and m_c and circumradius R, we have[4]^{: p.26, #954}

${\displaystyle m_{a}^{2}+m_{b}^{2}+m_{c}^{2}>6R^{2}}$

while the opposite inequality holds for an obtuse triangle.

Also, an acute triangle satisfies[4]^{: p.26, #954}

${\displaystyle r^{2}+r_{a}^{2}+r_{b}^{2}+r_{c}^{2}<8R^{2},}$

in terms of the excircle radii r_a , r_b , and r_c , again with the reverse inequality holding for an obtuse triangle.

For an acute triangle with semiperimeter s,[4]^{: p.115, #2874}

${\displaystyle s-r>2R,}$

and the reverse inequality holds for an obtuse triangle.

For an acute triangle with area K,[4]^{: p.185, #291.6}

${\displaystyle ab+bc+ca\geq 2R(R+r)+{\frac {8K}{\sqrt {3}}}.}$

For an acute triangle the distance between the circumcenter O and the orthocenter H satisfies[4]^{: p.26, #954}

${\displaystyle OH<R,}$

with the opposite inequality holding for an obtuse triangle.

For an acute triangle the distance between the incircle center I and orthocenter H satisfies[4]^{: p.26, #954}

${\displaystyle IH<r{\sqrt {2}},}$

where r is the inradius, with the reverse inequality for an obtuse triangle.

If one of the inscribed squares of an acute triangle has side length x_a and another has side length x_b with x_a < x_b, then[2]^{: p. 115}

${\displaystyle 1\geq {\frac {x_{a}}{x_{b}}}\geq {\frac {2{\sqrt {2}}}{3}}\approx 0.94.}$

If two obtuse triangles have sides (a, b, c) and (p, q, r) with c and r being the respective longest sides, then[4]^{: p.29, #1030}

${\displaystyle ap+bq<cr.}$

The Calabi triangle, which is the only non-equilateral triangle for which the largest square that fits in the interior can be positioned in any of three different ways, is obtuse and isosceles with base angles 39.1320261...° and third angle 101.7359477...°.

The equilateral triangle, with three 60° angles, is acute.

The Morley triangle, formed from any triangle by the intersections of its adjacent angle trisectors, is equilateral and hence acute.

The golden triangle is the isosceles triangle in which the ratio of the duplicated side to the base side equals the golden ratio. It is acute, with angles 36°, 72°, and 72°, making it the only triangle with angles in the proportions 1:2:2.[5]

The heptagonal triangle, with sides coinciding with a side, the shorter diagonal, and the longer diagonal of a regular heptagon, is obtuse, with angles ${\displaystyle \pi /7,2\pi /7,}$ and ${\displaystyle 4\pi /7.}$

The only triangle with consecutive integers for an altitude and the sides is acute, having sides (13,14,15) and altitude from side 14 equal to 12.

The smallest-perimeter triangle with integer sides in arithmetic progression, and the smallest-perimeter integer-sided triangle with distinct sides, is obtuse: namely the one with sides (2, 3, 4).

The only triangles with one angle being twice another and having integer sides in arithmetic progression are acute: namely, the (4,5,6) triangle and its multiples.[6]

There are no acute integer-sided triangles with area = perimeter, but there are three obtuse ones, having sides[7] (6,25,29), (7,15,20), and (9,10,17).

The smallest integer-sided triangle with three rational medians is acute, with sides[8] (68, 85, 87).

Heron triangles have integer sides and integer area. The oblique Heron triangle with the smallest perimeter is acute, with sides (6, 5, 5). The two oblique Heron triangles that share the smallest area are the acute one with sides (6, 5, 5) and the obtuse one with sides (8, 5, 5), the area of each being 12.