Exhaustive Guide to Circle Geometry and Trigonometry

Circle: A set of points equidistant from a common point called the center of the circle. Circles are named by their center point.
Center: The common point from which all points on a circle are equidistant.
Radius: A segment whose endpoints are the center and any point on the circle.
Chord: A segment whose endpoints are on the circle.
Diameter: A chord that contains the center of the circle. A chord through the center of a circle divides the circle into two semicircles.
Secant: A line that intersects a circle in $2$ places.
Tangent: A segment, line, or ray that intersects the circle at exactly one point, known as the point of tangency.
Regions of a Circle: There are three distinct regions relative to a circle: - Interior - Exterior - On the circle
Concentric Circles: Coplanar circles that share a common center. They do not intersect and resemble a bullseye pattern.
Intersection of Coplanar Circles: Two circles in the same plane can have $2$ , $1$ , or no points of intersection.
Common Tangents: A line, ray, or segment that is tangent to two different coplanar circles is referred to as a common tangent.
Circular Similarity: The fundamental relationship between each circle is that all circles are similar.

General Chord Properties: - A chord is a segment with endpoints on a circle. - A chord divides the circle into two arcs.
Theorem 1: In the same or congruent circles, two minor arcs are congruent if and only if (IFF) their corresponding chords are congruent.
Theorem 2: If one chord is a perpendicular bisector of another chord, the first chord is a diameter.
Theorem 3: If a diameter of a circle is perpendicular to a chord, then the diameter bisects the chord and its arc.
Theorem 4: In the same circle or congruent circles, two chords are congruent IFF they are equidistant from the center.
Tangency Theorem 1: In a plane, a line is tangent to a circle IFF the line is perpendicular to a radius of the circle at its endpoint on the circle.
Tangency Theorem 2: Tangent segments from a common external point are congruent.

Segments of Chords Theorem: If two chords intersect in the interior of a circle, then the product of the lengths of the segments of one chord is equal to the product of the lengths of the segments of the other chord.
Segments of Secants Theorem: If two secant segments share the same endpoint outside a circle, then the product of the lengths of one secant segment and its external segment equals the product of the lengths of the other secant segment and its external segment.
Segments of Secants and Tangents Theorem: If a secant segment and a tangent segment share an endpoint outside a circle, then the product of the lengths of the secant segment and its external segment equals the square of the length of the tangent segment.

General Equation: The equation of any circle where $(h, k)$ is the center and $r$ is the radius is defined as: - $(x-h)^2+(y-k)^2=r^2$

Tangent-Chord Angles: If a tangent and a chord intersect at a point on a circle, then the measure of each angle formed is half the measure of the intersected arc.
Angles Inside the Circle Theorem: If two chords intersect inside a circle, then the measure of each angle is half the sum of the intercepted arcs.
Angles Outside the Circle Theorem: If a tangent and a secant, two tangents, or two secants intersect outside a circle, then the measure of the angle formed is half the difference of the intercepted arcs. This is calculated as: - $rac{1}{2} \times (\text{Big arc} - \text{small arc})$

Definition of Trigonometry: The ratios of the reduced sides of similar right triangles. All right triangles with a given acute angle are similar to every other triangle with that same angle measure, meaning side lengths are in proportion.
Foundational Ratios: - Sine: Calculated as the opposite leg divided by the hypotenuse. - Cosine: Calculated as the adjacent leg divided by the hypotenuse. - Tangent: Calculated as the opposite leg divided by the adjacent leg.
SohCohToa: An acronym used to remember the definitions of sine, cosine, and tangent.
Numerical Consistency: Ratios simplify to the same value for all triangles with consistent angle measures (e.g., all $30-60-90$ triangles).
Calculator Application: Calculators store these unique trigonometric ratios in decimal form. These can be used to set up proportions to find missing side lengths when only one angle and one side are known.
Inverse Trigonometry: If the sine, cosine, or tangent (the ratios of sides) of an acute angle is known, inverse trigonometric functions can find the measure of the angle. This process "undoes" the original trigonometric function.

$30-60-90$ Triangles: - The ratio of the sides is $a : a\sqrt{3} : 2a$ . - The longest side (hypotenuse) is exactly double the shortest side.
$45-45-90$ Triangles: - The ratio of the sides is $a : a : a\sqrt{2}$ .

Application: These laws are used for non-right triangles specifically.
Law of Sines: Used when given Angle-Side-Angle (ASA), Angle-Angle-Side (AAS), or Side-Side-Angle (SSA). - $\frac{\sin(A)}{a}=\frac{\sin(B)}{b}=\frac{\sin(C)}{c}$
Law of Cosines: Used when given Side-Angle-Side (SAS) or Side-Side-Side (SSS), particularly when looking for an angle measure. - $a^2=b^2+c^2-2bc \cdot \cos(A)$ - $b^2=a^2+c^2-2ac \cdot \cos(B)$ - $c^2=a^2+b^2-2ab \cdot \cos(C)$

Angle of Elevation: The angle formed by a horizontal line and a line of sight to a point located above that horizontal line.
Angle of Depression: The angle formed by a horizontal line and a line of sight to a point located below that horizontal line.
Congruence Rule: Because horizontal lines are parallel, the angle of depression and the angle of elevation are congruent due to the alternate interior angle theorem.

Midsegment: A segment that connects the midpoints of two sides of a triangle.
Perpendicular Bisector: A segment, ray, line, or plane that is perpendicular to a segment at its midpoint.
Angle Bisector: A segment, ray, line, or plane that divides an angle into two congruent angles.
Median: A segment from a vertex to the midpoint of the opposite side.
Altitude: A segment from a vertex that is perpendicular to the opposite side. In some cases, the side must be extended to meet the altitude.
Equidistant: A point is defined as being equidistant from two figures if the point is an equal distance from each figure.
Midsegment Theorem: - The segment connecting the midpoints of two sides of a triangle is parallel to the third side and is exactly half as long as that third side. - Every triangle contains exactly three midsegments. - The three midsegments together create four congruent triangles inside the original triangle.