Triangle

• A triangle (symbol $△$) is the union of three line segments that are determined by three noncollinear points.

• All triangles are convex

• In a triangle, the sum of the measures of the interior angles is $180^{\circ }$

• The Triangle Inequality - The length of each side in a triangle is less than the sum of the lengths of the other sides.

• If $a$, $b$, $c$ are positive real numbers each of which is less than the sum of the other two, then you can build a triangle whose side lengths are $a$, $b$, and $c$.

Triangle
Area	$A=\frac{h_{b}b}{2}=\sqrt{p(p-a)(p-b)(p-c)}=rp=\frac{abc}{4R}$
Base	$b$
Height	$h_b$
Sides Length	$a,b,c$
Semiperimeter	$p=\frac{a+b+c}{2}$
Perimeter	$P=a+b+c$
Inradius	$r=\frac{A}{p}$
Circumradius	$R=\frac{abc}{4A}$

• The incircle (or inscribed circle) of a triangle is the largest circle that can be contained in the triangle; it touches (is tangent to) the three sides.
  • The center of the incircle is a triangle center called the triangle’s incenter
• The circumcircle (or circumscribed circle) of a triangle is a circle that passes through all three vertices.
  • The center of this circle is called the circumcenter of the triangle
  • Its radius is called the circumradius

Triangle Centers

	Intersect / Centered	Trilinear Coordinates
Angle bisectors and incircle	$I$ - Incenter	$1:1:1$
Medians	$G$ - Centroid	$bc:ca:ab$
Perpendicular bisectors & circumcircle	$O$ - Circumcenter	$\cos A:\cos B:\cos C$
Altitudes	$H$ - Orthocenter	$\sec A:\sec B:\sec C$

Triangle Classification

Triangles Classified by Congruent Sides

Scalene triangle

• Scalene triangle - None of Congruent Sides

Isosceles triangle

• Isosceles triangle - Two of Congruent Sides
  • In an isosceles triangle, the base angles are equal to each other.
  • If two angles in a triangle are equal in size, then the opposite sides are equal in length.
  • A triangle is isosceles if and only if two of its angles are equal to each other
  • Let $△ABC$ be an isosceles triangle, suppose $\overline{AB}=\overline{AC}$. then the altitude, median, and angle bisector from the vertex $A$ all coincide

Equilateral triangle

• Equilateral triangle - Three of Congruent Sides
  • In a triangle, all of whose sides are equal to each other, all the angles are also equal to each other.
  • Each angle of an equiangular triangle measures 60°

Equilateral triangle
Angle	$60^{\circ }=\frac{\pi }{3}$
Area	$A=\frac{\sqrt{3}}{4}{a}^2=\frac{3\sqrt{3}}{4}{R}^2$
Height	$h=\frac{\sqrt{3}}{2}a$
Sides Length	$a=b=c$
Permiter	$P=3a$
Inradius	$r=\frac{\sqrt{3}}{6}a=\frac{R}{2}=\frac{h}{3}$
Circumradius	$R=\frac{a}{\sqrt{3}}=\frac{2h}{3}$
Apothem	$\frac{h}{3}$

Triangles Classified by Angles

• Right triangle - One right angle

  • Pythagoras theorem - In a right-angled triangle, the square of the hypotenuse is equal to the sum of the squares of the lengths of the perpendiculars: $a^2+b^2=c^2$
  • The acute angles of a right triangle are complementary.
  • The area of a right triangle with legs of lengths $a$ and $b$ is given by $\frac{ab}{2}$.

Right Triangle
Area	$A=\frac{ab}{2}$
Legs	$a,b$
Hypotenuse	$c=\sqrt{a^2+b^2}$
Sides Length	$a,b,c$
Semiperimeter	$p=\frac{a+b+c}{2}$
Perimeter	$P=a+b+c$
Inradius	$r=$
Circumradius	$R=$

• Equiangular - All angles congruent
• Acute - All angles acute
• Obtuse - One obtuse angle

Line Segments

• A median of a triangle is a line segment joining a vertex to the midpoint of the opposite side, thus bisecting that side

• An angle bisector is a straight line drawn from the vertex of a triangle to its opposite side in such a way, that it divides the angle into two equal or congruent angles.

• An altitude of a triangle is a line segment through a vertex and perpendicular to a line containing the side opposite the vertex.

  • The three altitudes in a triangular triangle (or their continuations) meet at one point.

• The perpendicular bisector of a line segment is a line which meets the segment at its midpoint perpendicularly.

• The three perpendicular bisectors of the sides of a triangle are concurrent

• The three altitudes of a triangle are concurrent.

• The three medians of a triangle are concurrent at a point that is two-thirds the distance from any vertex to the midpoint of the opposite side.

Congruence $\cong$

• Triangle Congruence Theorem
  • SAS (side-angle-side): If two pairs of sides of two triangles are equal in length, and the included angles are equal in measurement, then the triangles are congruent.
  • SSS (side-side-side): If three pairs of sides of two triangles are equal in length, then the triangles are congruent.
  • ASA (angle-side-angle): If two pairs of angles of two triangles are equal in measurement, and the included sides are equal in length, then the triangles are congruent.
  • SSA (side-side-angle) If two triangles equal in size to two sides and the angle opposite the larger side, then the triangles are congruent.

Similarity $\sim$

• Given the triangles $△A^{\prime }{B}^{\prime }{C}^{\prime }$ and $△ABC$
  • If $\frac{\overline{AB}}{\overline{A^{\prime }{B}^{\prime }}}=\frac{\overline{BC}}{\overline{B^{\prime }{C}^{\prime }}}=\frac{\overline{AC}}{\overline{A^{\prime }{C}^{\prime }}}$ then $△ABC\sim △A^{\prime }{B}^{\prime }{C}^{\prime }$
  • If it holds that the angles are equal: $\sphericalangle A=\sphericalangle A^{\prime },\sphericalangle B=\sphericalangle B^{\prime },\sphericalangle C=\sphericalangle C^{\prime }$ then $△ABC\sim △A^{\prime }{B}^{\prime }{C}^{\prime }$