List of Postulates and Theorems

Postulates and Theorems

Basic Geometric Postulates

Reflexive Property: Any segment or angle is congruent to itself.
SSS (Side-Side-Side) Congruence: If there exists a correspondence between the vertices of two triangles such that three sides of one triangle are congruent to the corresponding sides of the other triangle, the two triangles are congruent.
SAS (Side-Angle-Side) Congruence: If there exists a correspondence between the vertices of two triangles such that two sides and the included angle of one triangle are congruent to the corresponding parts of the other triangle, the two triangles are congruent.
ASA (Angle-Side-Angle) Congruence: If there exists a correspondence between the vertices of two triangles such that two angles and the included side of one triangle are congruent to the corresponding parts of the other triangle, the two triangles are congruent.
Two Points Determine a Line: Two points determine a line (or ray or segment).
HL (Hypotenuse-Leg) Congruence: If there exists a correspondence between the vertices of two right triangles such that the hypotenuse and a leg of one triangle are congruent to the corresponding parts of the other triangle, the two right triangles are congruent.
Shortest Path: A line segment is the shortest path between two points.
Parallel Postulate: Through a point not on a line, there is exactly one parallel to the given line.
Three Noncollinear Points Determine a Plane:.
Line Intersecting a Plane: If a line intersects a plane not containing it, then the intersection is exactly one point.
Intersection of Two Planes: If two planes intersect, their intersection is exactly one line.
AAA (Angle-Angle-Angle) Similarity: If there exists a correspondence between the vertices of two triangles such that the three angles of one triangle are congruent to the corresponding angles of the other triangle, then the triangles are similar.

Circle Properties

Tangent Line: A tangent line is perpendicular to the radius drawn to the point of contact.
Tangent Condition: If a line is perpendicular to a radius at its outer endpoint, then it is tangent to the circle.
Circumference of a Circle: $Circumference = \pi \cdot diameter$
Area of a Rectangle: The area of a rectangle is equal to the product of the base and the height for that base.

Area Postulates

Area Existence: Every closed region has an area.
Congruent Figures and Area: If two closed figures are congruent, then their areas are equal.
Area of Union: If two closed regions intersect only along a common boundary, then the area of their union is equal to the sum of their individual areas.
Area of a Circle: $Area = \pi r^2$ , where $r$ is the radius.
Total Area of a Sphere: $Area = 4\pi r^2$ , where $r$ is the sphere’s radius.
Volume of a Right Rectangular Prism: The volume of a right rectangular prism is equal to the product of its length, its width, and its height.

Inequality Theorems

Law of Trichotomy: For any two real numbers $x$ and $y$ , exactly one of the following statements is true: $x < y$ , $x = y$ , or $x > y$ .
Transitive Property of Inequality: If a > b and b > c, then a > c. Similarly, if x < y and y < z, then x < z.
Addition Property of Inequality: If a > b, then a + x > b + x.
Positive Multiplication Property of Inequality: If $x < y$ and $a > 0$ , then a \cdot x < a \cdot y.
Negative Multiplication Property of Inequality: If $x < y$ and $a < 0$ , then $a \cdot x > a \cdot y$ .
Triangle Inequality Theorem: The sum of the measures of any two sides of a triangle is always greater than the measure of the third side.

Congruence Theorems

Right Angles Congruence: If two angles are right angles, then they are congruent.
Straight Angles Congruence: If two angles are straight angles, then they are congruent.
Conditional and Contrapositive Statements: If a conditional statement is true, then the contrapositive of the statement is also true. (If $p$ , then $q$ . If $\sim q$ , then $\sim p$ .)

Supplementary and Complementary Angle Theorems

Supplementary Angles Theorem: If angles are supplementary to the same angle, then they are congruent.
Supplementary to Congruent Angles Theorem: If angles are supplementary to congruent angles, then they are congruent.
Complementary Angles Theorem: If angles are complementary to the same angle, then they are congruent.
Complementary to Congruent Angles Theorem: If angles are complementary to congruent angles, then they are congruent.

Addition and Subtraction Properties of Congruence

Addition Property (Segments): If a segment is added to two congruent segments, the sums are congruent.
Addition Property (Angles): If an angle is added to two congruent angles, the sums are congruent.
Addition Property (Congruent Segments): If congruent segments are added to congruent segments, the sums are congruent.
Addition Property (Congruent Angles): If congruent angles are added to congruent angles, the sums are congruent.
Subtraction Property (Segments/Angles): If a segment (or angle) is subtracted from congruent segments (or angles), the differences are congruent.
Subtraction Property (Congruent Segments/Angles): If congruent segments (or angles) are subtracted from congruent segments (or angles), the differences are congruent.

Multiplication and Division Properties of Congruence

Multiplication Property: If segments (or angles) are congruent, their like multiples are congruent.
Division Property: If segments (or angles) are congruent, their like divisions are congruent.

Transitive Property of Congruence

Transitive Property: If angles (or segments) are congruent to the same angle (or segment), they are congruent to each other.
Transitive Property: If angles (or segments) are congruent to congruent angles (or segments), they are congruent to each other.

Triangle Properties

Vertical Angles: Vertical angles are congruent.
Radii of a Circle: All radii of a circle are congruent.
Isosceles Triangle Theorem: If two sides of a triangle are congruent, the angles opposite the sides are congruent. (If $Side1 = Side2$ , then $Angle1 = Angle2$ .)
Converse of Isosceles Triangle Theorem: If two angles of a triangle are congruent, the sides opposite the angles are congruent. (If $Angle1 = Angle2$ , then $Side1 = Side2$ .)
Midpoint Formula: If $A = (x1, y1)$ and $B = (x2, y2)$ , then the midpoint $M = (xm, ym)$ of $AB$ can be found by using the midpoint formula: $M = (xm, ym) = (\frac{x1+x2}{2}, \frac{y1+y2}{2})$ .
Supplementary and Congruent Angles: If two angles are both supplementary and congruent, then they are right angles.

Perpendicular Bisector Theorems

Perpendicular Bisector Determination: If two points are each equidistant from the endpoints of a segment, then the two points determine the perpendicular bisector of that segment.
Point on Perpendicular Bisector: If a point is on the perpendicular bisector of a segment, then it is equidistant from the endpoints of that segment.

Parallel and Perpendicular Lines

Parallel Lines and Slopes: If two nonvertical lines are parallel, then their slopes are equal.
Equal Slopes and Parallel Lines: If the slopes of two nonvertical lines are equal, then the lines are parallel.
Perpendicular Lines and Slopes: If two lines are perpendicular and neither is vertical, each line’s slope is the opposite reciprocal of the other’s.
Opposite Reciprocal Slopes and Perpendicular Lines: If a line’s slope is the opposite reciprocal of another line’s slope, the two lines are perpendicular.
Exterior Angle Inequality Theorem: The measure of an exterior angle of a triangle is greater than the measure of either remote interior angle.

Transversal Theorems

Alternate Interior Angles Converse: If two lines are cut by a transversal such that two alternate interior angles are congruent, the lines are parallel. (Alt. int. $\angle$ s = $\parallel$ lines)
Alternate Exterior Angles Converse: If two lines are cut by a transversal such that two alternate exterior angles are congruent, the lines are parallel. (Alt. ext. $\angle$ s = $\parallel$ lines)
Corresponding Angles Converse: If two lines are cut by a transversal such that two corresponding angles are congruent, the lines are parallel. (Corr. $\angle$ s = $\parallel$ lines)
Same-Side Interior Angles Supplementary Converse: If two lines are cut by a transversal such that two interior angles on the same side of the transversal are supplementary, the lines are parallel.
Same-Side Exterior Angles Supplementary Converse: If two lines are cut by a transversal such that two exterior angles on the same side of the transversal are supplementary, the lines are parallel.
Lines Perpendicular to a Third Line: If two coplanar lines are perpendicular to a third line, they are parallel.
Parallel Lines and Alternate Interior Angles: If two parallel lines are cut by a transversal, each pair of alternate interior angles are congruent. ( $\parallel$ lines $\implies$ alt. int. $\angle$ s =)
Parallel Lines and Angle Pairs: If two parallel lines are cut by a transversal, then any pair of the angles formed are either congruent or supplementary.
Parallel Lines and Alternate Exterior Angles: If two parallel lines are cut by a transversal, each pair of alternate exterior angles are congruent. ( $\parallel$ lines $\implies$ alt. ext. $\angle$ s =)
Parallel Lines and Corresponding Angles: If two parallel lines are cut by a transversal, each pair of corresponding angles are congruent. ( $\parallel$ lines $\implies$ corr. $\angle$ s =)
Parallel Lines and Same-Side Interior Angles: If two parallel lines are cut by a transversal, each pair of interior angles on the same side of the transversal are supplementary.
Parallel Lines and Same-Side Exterior Angles: If two parallel lines are cut by a transversal, each pair of exterior angles on the same side of the transversal are supplementary.
Perpendicular to Parallel Lines: In a plane, if a line is perpendicular to one of two parallel lines, it is perpendicular to the other.
Transitive Property of Parallel Lines: If two lines are parallel to a third line, they are parallel to each other.

Plane Determination

Line and a Point: A line and a point not on the line determine a plane.
Intersecting Lines: Two intersecting lines determine a plane.
Parallel Lines: Two parallel lines determine a plane.

Line Perpendicular to a Plane

Perpendicularity Condition: If a line is perpendicular to two distinct lines that lie in a plane and that pass through its foot, then it is perpendicular to the plane.
Parallel Planes and Intersection: If a plane intersects two parallel planes, the lines of intersection are parallel.

Triangle Angle Sum Theorem

Sum of Angles in a Triangle: The sum of the measures of the three angles of a triangle is 180.
Exterior Angle Theorem: The measure of an exterior angle of a triangle is equal to the sum of the measures of the remote interior angles.

Midline Theorem

Midline Theorem: A segment joining the midpoints of two sides of a triangle is parallel to the third side, and its length is one-half the length of the third side.

No-Choice Theorem

No-Choice Theorem: If two angles of one triangle are congruent to two angles of a second triangle, then the third angles are congruent.

AAS Congruence

AAS (Angle-Angle-Side) Congruence: If there exists a correspondence between the vertices of two triangles such that two angles and a nonincluded side of one are congruent to the corresponding parts of the other, then the triangles are congruent.

Polygon Angle Sum

Interior Angle Sum: The sum $Si$ of the measures of the angles of a polygon with $n$ sides is given by the formula $Si = (n - 2)180$ .
Exterior Angle Sum: If one exterior angle is taken at each vertex, the sum of the measures of the exterior angles of a polygon is given by the formula $S_e = 360$ .
Number of Diagonals: The number $d$ of diagonals that can be drawn in a polygon of $n$ sides is given by the formula $d = \frac{n(n - 3)}{2}$ .
Exterior Angle of Equiangular Polygon: The measure $E$ of each exterior angle of an equiangular polygon of $n$ sides is given by the formula $E = \frac{360}{n}$ .

Proportion Theorems

Means-Extremes Products Theorem: In a proportion, the product of the means is equal to the product of the extremes.
Means-Extremes Ratio Theorem: If the product of a pair of nonzero numbers is equal to the product of another pair of nonzero numbers, then either pair of numbers may be made the extremes, and the other pair the means, of a proportion.

Similar Polygon Perimeters

Perimeter Ratio: The ratio of the perimeters of two similar polygons equals the ratio of any pair of corresponding sides.

Triangle Similarity Theorems

AA (Angle-Angle) Similarity: If there exists a correspondence between the vertices of two triangles such that two angles of one triangle are congruent to the corresponding angles of the other, then the triangles are similar.
SSS (Side-Side-Side) Similarity: If there exists a correspondence between the vertices of two triangles such that the ratios of the measures of corresponding sides are equal, then the triangles are similar.
SAS (Side-Angle-Side) Similarity: If there exists a correspondence between the vertices of two triangles such that the ratios of the measures of two pairs of corresponding sides are equal and the included angles are congruent, then the triangles are similar.

Side-Splitter Theorem

Side-Splitter Theorem: If a line is parallel to one side of a triangle and intersects the other two sides, it divides those two sides proportionally.
Parallel Lines and Transversals: If three or more parallel lines are intersected by two transversals, the parallel lines divide the transversals proportionally.

Angle Bisector Theorem

Angle Bisector Theorem: If a ray bisects an angle of a triangle, it divides the opposite side into segments that are proportional to the adjacent sides.

Right Triangle Similarity

Altitude to Hypotenuse Theorem: If an altitude is drawn to the hypotenuse of a right triangle, then:
- The two triangles formed are similar to the given right triangle and to each other.
- The altitude to the hypotenuse is the mean proportional between the segments of the hypotenuse.
- Either leg of the given right triangle is the mean proportional between the hypotenuse of the given right triangle and the segment of the hypotenuse adjacent to that leg (i.e., the projection of that leg on the hypotenuse).

Pythagorean Theorem

Pythagorean Theorem: The square of the measure of the hypotenuse of a right triangle is equal to the sum of the squares of the measures of the legs.
Converse of Pythagorean Theorem: If the square of the measure of one side of a triangle equals the sum of the squares of the measures of the other two sides, then the angle opposite the longest side is a right angle.

Distance Formula

Distance Formula: If $P = (x1, y1)$ and $Q = (x2, y2)$ are any two points, then the distance between them can be found with the formula $PQ = \sqrt{(x2 - x1)^2 + (y2 - y1)^2}$ or $PQ = \sqrt{(\Delta x)^2 + (\Delta y)^2}$ .

Special Right Triangles

30-60-90 Triangle Theorem: In a triangle whose angles have the measures 30, 60, and 90, the lengths of the sides opposite these angles can be represented by $x$ , $x\sqrt{3}$ , and $2x$ respectively.
45-45-90 Triangle Theorem: In a triangle whose angles have the measures 45, 45, and 90, the lengths of the sides opposite these angles can be represented by $x$ , $x$ , and $x\sqrt{2}$ respectively.

Circle and Chord Relationships

Radius Perpendicular to Chord: If a radius is perpendicular to a chord, then it bisects the chord.
Radius Bisecting Chord: If a radius of a circle bisects a chord that is not a diameter, then it is perpendicular to that chord.
Perpendicular Bisector of Chord: The perpendicular bisector of a chord passes through the center of the circle.
Equidistant Chords: If two chords of a circle are equidistant from the center, then they are congruent.
Congruent Chords: If two chords of a circle are congruent, then they are equidistant from the center of the circle.

Central Angles and Arcs

Congruent Central Angles and Arcs: If two central angles of a circle (or of congruent circles) are congruent, then their intercepted arcs are congruent.
Congruent Arcs and Central Angles: If two arcs of a circle (or of congruent circles) are congruent, then the corresponding central angles are congruent.
Congruent Central Angles and Chords: If two central angles of a circle (or of congruent circles) are congruent, then the corresponding chords are congruent.
Congruent Chords and Central Angles: If two chords of a circle (or of congruent circles) are congruent, then the corresponding central angles are congruent.
Congruent Arcs and Chords: If two arcs of a circle (or of congruent circles) are congruent, then the corresponding chords are congruent.
Congruent Chords and Arcs: If two chords of a circle (or of congruent circles) are congruent, then the corresponding arcs are congruent.

Tangent Theorems

Two-Tangent Theorem: If two tangent segments are drawn to a circle from an external point, then those segments are congruent.

Angle and Arc Relationships

Inscribed Angle Theorem: The measure of an inscribed angle or a tangent-chord angle (vertex on a circle) is one-half the measure of its intercepted arc.
Chord-Chord Angle Theorem: The measure of a chord-chord angle is one-half the sum of the measures of the arcs intercepted by the chord-chord angle and its vertical angle.
Secant-Secant/Secant-Tangent/Tangent-Tangent Angle Theorem: The measure of a secant-secant angle, a secant-tangent angle, or a tangent-tangent angle (vertex outside a circle) is one-half the difference of the measures of the intercepted arcs.
Inscribed/Tangent-Chord Angles and Same Arc: If two inscribed or tangent-chord angles intercept the same arc, then they are congruent.
Inscribed/Tangent-Chord Angles and Congruent Arcs: If two inscribed or tangent-chord angles intercept congruent arcs, then they are congruent.
Angle Inscribed in a Semicircle: An angle inscribed in a semicircle is a right angle.
Tangent-Tangent Angle and Minor Arc: The sum of the measures of a tangent-tangent angle and its minor arc is 180.
Inscribed Quadrilateral: If a quadrilateral is inscribed in a circle, its opposite angles are supplementary.
Inscribed Parallelogram: If a parallelogram is inscribed in a circle, it must be a rectangle.

Chord Segment Theorem

Chord-Chord Power Theorem: If two chords of a circle intersect inside the circle, then the product of the measures of the segments of one chord is equal to the product of the measures of the segments of the other chord.

Tangent-Secant Theorem

Tangent-Secant Power Theorem: If a tangent segment and a secant segment are drawn from an external point to a circle, then the square of the measure of the tangent segment is equal to the product of the measures of the entire secant segment and its external part.

Secant Theorem

Secant-Secant Power Theorem: If two secant segments are drawn from an external point to a circle, then the product of the measures of one secant segment and its external part is equal to the product of the measures of the other secant segment and its external part.

Arc Length

Arc Length Formula: The length of an arc is equal to the circumference of its circle times the fractional part of the circle determined by the arc.

Area Formulas

Area of a Square: The area of a square is equal to the square of a side.
Area of a Parallelogram: The area of a parallelogram is equal to the product of the base and the height.
Area of a Triangle: The area of a triangle is equal to one-half the product of a base and the height (or altitude) for that base.
Area of a Trapezoid: The area of a trapezoid equals one-half the product of the height and the sum of the bases.
Median of a Trapezoid: The measure of the median of a trapezoid equals the average of the measures of the bases.
Area of a Trapezoid (Median): The area of a trapezoid is the product of the median and the height.
Area of a Kite: The area of a kite equals half the product of its diagonals.
Area of an Equilateral Triangle: The area of an equilateral triangle equals the product of one-fourth the square of a side and the square root of 3.
Area of a Regular Polygon: The area of a regular polygon equals one-half the product of the apothem and the perimeter.
Area of a Sector: The area of a sector of a circle is equal to the area of the circle times the fractional part of the circle determined by the sector’s arc.

Similar Figures Theorem

Ratio of Areas: If two figures are similar, then the ratio of their areas equals the square of the ratio of corresponding segments. (Similar-Figures Theorem)

Triangle Medians and Area

Median and Triangle Area: A median of a triangle divides the triangle into two triangles with equal areas.

Hero's Formula

Area of a Triangle (Hero’s Formula): $Area = \sqrt{s(s - a)(s - b)(s - c)}$ , where $a$ , $b$ , and $c$ are the lengths of the sides of the triangle and $s = semiperimeter = \frac{a + b + c}{2}$ .

Brahmagupta's Formula

Area of a Cyclic Quadrilateral (Brahmagupta’s Formula): $Area = \sqrt{(s - a)(s - b)(s - c)(s - d)}$ , where $a$ , $b$ , $c$ , and $d$ are the sides of the quadrilateral and $s = semiperimeter = \frac{a + b + c + d}{2}$ .

Lateral Area Formulas

Lateral Area of a Cylinder: The lateral area of a cylinder is equal to the product of the height and the circumference of the base.
Lateral Area of a Cone: The lateral area of a cone is equal to one-half the product of the slant height and the circumference of the base.

Volume Formulas

Volume of a Right Rectangular Prism: The volume of a right rectangular prism is equal to the product of the height and the area of the base.
Volume of any Prism: The volume of any prism is equal to the product of the height and the area of the base.
Volume of a Cylinder: The volume of a cylinder is equal to the product of the height and the area of the base.
Volume of a Prism/Cylinder (Cross-Sectional Area): The volume of a prism or a cylinder is equal to the product of the figure’s cross-sectional area and its height.
Volume of a Pyramid: The volume of a pyramid is equal to one third of the product of the height and the area of the base.
Volume of a Cone: The volume of a cone is equal to one third of the product of the height and the area of the base.

Pyramid/Cone Section Theorem

Ratio of Cross-Sectional Area: In a pyramid or a cone, the ratio of the area of a cross section to the area of the base equals the square of the ratio of the figures’ respective distances from the vertex.

Sphere Volume

Volume of a Sphere: The volume of a sphere is equal to four thirds of the product of $\pi$ and the cube of the radius.

Linear Equations

Slope-Intercept Form: The y-form, or slope-intercept form, of the equation of a nonvertical line is $y = mx + b$ , where $b$ is the y-intercept of the line and $m$ is the slope of the line.
Horizontal Line: The formula for an equation of a horizontal line is $y = b$ , where $b$ is the y-coordinate of every point on the line.
Vertical Line: The formula for the equation of a vertical line is $x = a$ , where $a$ is the x-coordinate of every point on the line.

3D Distance Formula

3D Distance Formula: If $P = (x1, y1, z1)$ and $Q = (x2, y2, z2)$ are any two points, then the distance between them can be found with the formula $PQ = \sqrt{(x2 - x1)^2 + (y2 - y1)^2 + (z2 - z1)^2}$ .

Circle Equation

Equation of a Circle: The equation of a circle whose center is $(h, k)$ and whose radius is $r$ is $(x - h)^2 + (y - k)^2 = r^2$ .

Concurrency Theorems

Circumcenter Theorem: The perpendicular bisectors of the sides of a triangle are concurrent at a point that is equidistant from the vertices of the triangle. (The point of concurrency of the perpendicular bisectors is called the circumcenter of the triangle.)
Incenter Theorem: The bisectors of the angles of a triangle are concurrent at a point that is equidistant from the sides of the triangle. (The point of concurrency of the angle bisectors is called the incenter of the triangle.)
Orthocenter Theorem: The lines containing the altitudes of a triangle are concurrent. (The point of concurrency of the lines containing the altitudes is called the orthocenter of the triangle.)
Centroid Theorem: The medians of a triangle are concurrent at a point that is two thirds of the way from any vertex of the triangle to the midpoint of the opposite side. (The point of concurrency of the medians of a triangle is called the centroid of the triangle.)

Side-Angle Inequality Theorems

Unequal Sides and Angles: If two sides of a triangle are not congruent, then the angles opposite them are not congruent, and the larger angle is opposite the longer side. (If $Side1 > Side2$ , then $Angle1 > Angle2$ .)
Unequal Angles and Sides: If two angles of a triangle are not congruent, then the sides opposite them are not congruent, and the longer side is opposite the larger angle. (If $Angle1 > Angle2$ , then $Side1 > Side2$ .)
SAS Inequality Theorem: If two sides of one triangle are congruent to two sides of another triangle and the included angle in the first triangle is greater than the included angle in the second triangle, then the remaining side of the first triangle is greater than the remaining side of the second triangle. (SAS Inequality)
SSS Inequality Theorem: If two sides of one triangle are congruent to two sides of another triangle and the third side of the first triangle is greater than the third side of the second triangle, then the angle opposite the third side in the first triangle is greater than the angle opposite the third side in the second triangle. (SSS Inequality)

Point-to-Line Distance

Distance from a Point to a Line: The distance $d$ from any point $P = (x1, y1)$ to a line whose equation is in the form $ax + by + c = 0$ can be found with the formula $d = \frac{|ax1 + by1 + c|}{\sqrt{a^2 + b^2}}$ .

Triangle Area with Coordinates

Area of a Triangle (Coordinates): The area $A$ of a triangle with vertices at $(x1, y1)$ , $(x2, y2)$ , and $(x3, y3)$ can be found with the formula $A = \frac{1}{2} |x1y2 + x2y3 + x3y1 - x1y3 - x2y1 - x3y2|$ .

Law of Sines

Law of Sines: In any triangle ABC, with side lengths $a$ , $b$ , and $c$ , $\frac{a}{\sin \angle A} = \frac{b}{\sin \angle B} = \frac{c}{\sin \angle C} = D$ , where $D$ is the diameter of the triangle's circumcircle.

Stewart's Theorem

Stewart’s Theorem: In any triangle ABC, with side lengths a, b, and c,
$a^2n + b^2m = c(d^2 + mn)$
where d is the length of a segment from vertex C to the opposite side, dividing that side into segments with lengths m and n.

Ptolemy's Theorem

Ptolemy's Theorem: If a quadrilateral is inscribable in a circle, the product of the measures of its diagonals is equal to the sum of the products of the measures of the pairs of opposite sides.

Inradius of a Triangle

Inradius Formula: The inradius r of a triangle can be found with the formula $r = \frac{A}{s}$ , where A is the triangle’s area and s is the triangle's semiperimeter.

Circumradius of a Triangle

Circumradius Formula: The circumradius R of a triangle can be found with the formula $R = \frac{abc}{4A}$ , where a, b, and c are the lengths of the sides of the triangle and A is the triangle’s area.

Ceva's Theorem

Ceva's Theorem: If ABC is a triangle with D on BC, E on AC, and F on AB, then the three segments AD, BE, and CF are concurrent if, and only if, $\frac{BD}{DC} \cdot \frac{CE}{EA} \cdot \frac{AF}{FB} = 1$

Menelaus' Theorem

Menelaus' Theorem: If ABC is a triangle and F is on AB, E is on AC, and D is on an extension of BC, then the three points D, E, and F are collinear if, and only if, $\frac{BD}{DC} \cdot \frac{CE}{EA} \cdot \frac{AF}{FB} = -1$