Similar Triangles:
- Corresponding angles are equal.
- Corresponding sides are in the same ratio.
Similarity Criteria: AAA, SSS, SAS.
Basic Proportionality Theorem (Thales Theorem): If a line is drawn parallel to one side of a triangle to intersect the other two sides in distinct points, the other two sides are divided in the same ratio.
Converse of BPT: If a line divides any two sides of a triangle in the same ratio, then the line is parallel to the third side.
Areas of Similar Triangles: Ratio of areas is equal to the square of the ratio of their corresponding sides. $\text{Area}(\triangle ABC) / \text{Area}(\triangle PQR) = (AB/PQ)^2 = (BC/QR)^2 = (CA/RP)^2$.
Pythagoras Theorem: In a right triangle, the square of the hypotenuse is equal to the sum of the squares of the other two sides ($h^2 = p^2 + b^2$).
Converse of Pythagoras Theorem: If in a triangle, the square of one side is equal to the sum of the squares of the other two sides, then the angle opposite the first side is a right angle.

Distance Formula: Distance between $(x_1, y_1)$ and $(x_2, y_2)$ is $\sqrt{(x_2-x_1)^2 + (y_2-y_1)^2}$.
Section Formula: Point $P(x,y)$ dividing line segment joining $A(x_1, y_1)$ and $B(x_2, y_2)$ internally in ratio $m:n$ is: $x = (mx_2 + nx_1) / (m+n)$, $y = (my_2 + ny_1) / (m+n)$.
Mid-point Formula: $( (x_1+x_2)/2, (y_1+y_2)/2 )$.
Area of a Triangle: Vertices $(x_1, y_1), (x_2, y_2), (x_3, y_3)$: $1/2 |x_1(y_2-y_3) + x_2(y_3-y_1) + x_3(y_1-y_2)|$. If area is 0, points are collinear.

Trigonometric Ratios (Right Triangle):
- $\sin A = \text{Opposite} / \text{Hypotenuse}$
- $\cos A = \text{Adjacent} / \text{Hypotenuse}$
- $\tan A = \text{Opposite} / \text{Adjacent}$
- $\csc A = 1/\sin A$
- $\sec A = 1/\cos A$
- $\cot A = 1/\tan A$
Identities:
- $\sin^2 A + \cos^2 A = 1$
- $1 + \tan^2 A = \sec^2 A$
- $1 + \cot^2 A = \csc^2 A$
Complementary Angles:
- $\sin(90^\circ - A) = \cos A$
- $\cos(90^\circ - A) = \sin A$
- $\tan(90^\circ - A) = \cot A$
- $\cot(90^\circ - A) = \tan A$
- $\sec(90^\circ - A) = \csc A$
- $\csc(90^\circ - A) = \sec A$

Trigonometric Table (Common Angles):

Angle $\theta$	$0^\circ$	$30^\circ$	$45^\circ$	$60^\circ$	$90^\circ$
$\sin \theta$	$0$	$1/2$	$1/\sqrt{2}$	$\sqrt{3}/2$	$1$
$\cos \theta$	$1$	$\sqrt{3}/2$	$1/\sqrt{2}$	$1/2$	$0$