# Scalene Triangle

Sides of a triangle: $$a$$, $$b$$, $$c$$
Angles of a triangle: $$A = \alpha,$$ $$B = \beta,$$ $$C = \gamma$$
Altitudes to the sides $$a$$, $$b$$, $$c$$: $${h_a},$$ $${h_b},$$ $${h_c}$$
Medians to the sides $$a$$, $$b$$, $$c$$: $${m_a},$$ $${m_b},$$ $${m_c}$$
Bisectors of the angles $$\alpha$$, $$\beta$$, $$\gamma$$: $${t_a},$$ $${t_b},$$ $${t_c}$$
Midline of a triangle: $$q$$
Radius of circumscribed circle: $$R$$
Radius of inscribed circle: $$r$$
Semiperimeter of a triangle: $$p$$ Area: $$S$$
1. A triangle is a geometric shape formed by three line segments that join three points not lying on a single line.
2. Sum of the angles in a triangle is $$180^\circ$$:
$$\alpha + \beta + \gamma = 180^\circ$$
3. Triangle inequality
$$a + b \gt c$$
$$b + c \gt a$$
$$a + c \gt b$$
4. $$\left| {a – b} \right| \lt c$$
$$\left| {b – c} \right| \lt a$$
$$\left| {a – c} \right| \lt b$$
5. Midline of a triangle
$$q = a/2,\;\;q\parallel a$$
6. Law of Cosines
$${a^2} = {b^2} + {c^2}$$ $$-\; 2bc\cos \alpha$$
$${b^2} = {a^2} + {c^2}$$ $$-\; 2ac\cos \beta$$
$${c^2} = {a^2} + {b^2}$$ $$-\; 2ab\cos \gamma$$
7. Law of Sines
$${\large\frac{a}{{\sin \alpha }}\normalsize} = {\large\frac{b}{{\sin \beta }}\normalsize} =$$ $${\large\frac{c}{{\sin \gamma }}\normalsize}$$ $$= 2R,$$
where $$R$$ is the radius of the circumscribed circle.
8. Radius of the circumscribed circle
$$R = {\large\frac{a}{{\sin \alpha }}\normalsize} = {\large\frac{b}{{\sin \beta }}\normalsize} =$$ $${\large\frac{c}{{\sin \gamma }}\normalsize} =$$ $${\large\frac{{bc}}{{2{h_a}}}\normalsize} =$$ $${\large\frac{{ac}}{{2{h_b}}}\normalsize} =$$ $${\large\frac{{ab}}{{2{h_c}}}\normalsize} =$$ $${\large\frac{{abc}}{{4S}}\normalsize}$$
9. Radius of the inscribed circle
$${r^2} = {\large\frac{{\left( {p – a} \right)\left( {p – b} \right)\left( {p – c} \right)}}{p}\normalsize},$$ $${\large\frac{1}{r}\normalsize} = {\large\frac{1}{{{h_a}}}\normalsize} + {\large\frac{1}{{{h_b}}}\normalsize} + {\large\frac{1}{{{h_c}}}\normalsize}$$
10. Finding the angles of a triangle given $$3$$ sides
$$\sin {\large\frac{\alpha }{2}\normalsize} = \sqrt {\large\frac{{\left( {p – a} \right)\left( {p – c} \right)}}{{bc}}\normalsize}$$
$$\cos{\large\frac{\alpha }{2}\normalsize} = \sqrt {\large\frac{{p\left( {p – a} \right)}}{{bc}}\normalsize}$$
$$\tan{\large\frac{\alpha }{2}\normalsize} = \sqrt {\large\frac{{\left( {p – b} \right)\left( {p – c} \right)}}{{p\left( {p – a} \right)}}\normalsize}$$
11. Finding the altitudes of a triangle given $$3$$ sides
$${h_a} =$$ $${\large\frac{2}{a}\normalsize}\sqrt {p\left( {p – a} \right)\left( {p – b} \right)\left( {p – c} \right)}$$
$${h_b} =$$ $${\large\frac{2}{b}\normalsize}\sqrt {p\left( {p – a} \right)\left( {p – b} \right)\left( {p – c} \right)}$$
$${h_c} =$$ $${\large\frac{2}{c}\normalsize}\sqrt {p\left( {p – a} \right)\left( {p – b} \right)\left( {p – c} \right)}$$
12. Finding the altitudes of a triangle given a side and an angle
$${h_a} = b\sin \gamma = c\sin \beta$$
$${h_b} = a\sin \gamma = c\sin \alpha$$
$${h_c} = a\sin \beta = b\sin \alpha$$
13. Finding the medians of a triangle given $$3$$ sides
$$m_a^2 = {\large\frac{{{b^2} + {c^2}}}{2}\normalsize} – {\large\frac{{{a^2}}}{4}\normalsize}$$
$$m_b^2 = {\large\frac{{{a^2} + {c^2}}}{2}\normalsize} – {\large\frac{{{b^2}}}{4}\normalsize}$$
$$m_c^2 = {\large\frac{{{a^2} + {b^2}}}{2}\normalsize} – {\large\frac{{{c^2}}}{4}\normalsize}$$
14. Distance from a vertex to the point of intersection of the medians (centroid)
$$AM = {\large\frac{2}{3}\normalsize}{m_a},\;$$ $$BM = {\large\frac{2}{3}\normalsize}{m_b},\;$$ $$CM = {\large\frac{2}{3}\normalsize}{m_c}$$
15. Finding the bisectors given 3 sides
$$t_a^2 = {\large\frac{{4bcp\left( {p – a} \right)}}{{{{\left( {b + c} \right)}^2}}}\normalsize}$$
$$t_b^2 = {\large\frac{{4acp\left( {p – b} \right)}}{{{{\left( {a + c} \right)}^2}}}\normalsize}$$
$$t_c^2 = {\large\frac{{4abp\left( {p – c} \right)}}{{{{\left( {a + b} \right)}^2}}}\normalsize}$$
16. Area of a triangle
$$S = {\large\frac{{a{h_a}}}{2}\normalsize} = {\large\frac{{b{h_b}}}{2}\normalsize} = {\large\frac{{c{h_c}}}{2}\normalsize}$$
$$S = {\large\frac{{ab\sin \gamma }}{2}\normalsize} = {\large\frac{{ac\sin \beta }}{2}\normalsize} =$$ $${\large\frac{{bc\sin \alpha }}{2}\normalsize}$$
$$S = pr$$
$$S = {\large\frac{{abc}}{{4R}}\normalsize}$$
$$S = 2{R^2}\sin \alpha \sin \beta \sin \gamma$$
$$S = {p^2}\tan{\large\frac{\alpha }{2}\normalsize} \tan{\large\frac{\beta }{2}\normalsize} \tan{\large\frac{\gamma }{2}\normalsize}$$
17. Heron’s formula
$$S =$$ $$\sqrt {p\left( {p – a} \right)\left( {p – b} \right)\left( {p – c} \right)}$$
where $$p$$ is the semiperimeter of the triangle.