Triangles

At a glance 14 formulas

1. 1Third side$|b - c| < a < b + c$$a, b, c = \text{sides; any two exceed the third}$
2. 2Acute, right or obtuse$c^2 < a^2 + b^2:\ \text{acute}$$c^2 > a^2 + b^2:\ \text{obtuse};\ c = \text{longest side}$
3. 3Foot of an altitude$BM = \dfrac{c^2 + a^2 - b^2}{2a}$$M = \text{foot from } A;\ a, b, c \text{ opposite } A, B, C$
4. 4Pythagorean triplets

   3, 4

   5

   5, 12

   13

   8, 15

   17

   7, 24

   25

   20, 21

   29

   9, 40

   41

5. 5Area$\tfrac{1}{2} \times \text{base} \times \text{height} = \tfrac{1}{2}ab\sin C$$C = \text{angle between sides } a \text{ and } b$
6. 6Heron's formula$\text{area} = \sqrt{s(s - a)(s - b)(s - c)}$$a, b, c = \text{sides},\ s = \tfrac{a + b + c}{2}$
7. 7Equilateral, side a$\text{height} = \tfrac{\sqrt{3}}{2}a,\ \text{area} = \tfrac{\sqrt{3}}{4}a^2$$\text{inradius} = \tfrac{a}{2\sqrt{3}},\ \text{circumradius} = \tfrac{a}{\sqrt{3}}$
8. 8Inradius r, circumradius R$r = \dfrac{\text{area}}{s},\ R = \dfrac{abc}{4 \times \text{area}}$$s = \tfrac{a + b + c}{2};\ \text{hyp. } c:\ R = \tfrac{c}{2},\ r = \tfrac{a + b - c}{2}$
9. 9Altitude to hypotenuse$h = \dfrac{ab}{c}$$a, b = \text{legs},\ c = \text{hypotenuse}$
10. 10Similar triangles$\text{sides in ratio } k \Rightarrow \text{areas in } k^2$$DE \parallel BC:\ \tfrac{AD}{DB} = \tfrac{AE}{EC};\ D, E \text{ on } AB, AC$
11. 11Same heightTriangles with a common apex and bases on one line have areas in the ratio of their bases, so a median cuts a triangle into two equal halves.
12. 12Angle bisector theorem$\dfrac{BD}{DC} = \dfrac{AB}{AC}$$AD \text{ bisects } \angle A,\ D \text{ on } BC$
13. 13Median AD, centroid G$AB^2 + AC^2 = 2(AD^2 + BD^2)$$D = \text{midpoint of } BC;\ AG : GD = 2 : 1$
14. 14Median as half sideIf the median from A is half of BC, the midpoint of BC is equally far from A, B and C, so $\angle A = 90^\circ$. The converse holds too.