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Sine Theorem

Lecture






Theorem

The sides of the triangle are proportional to the sines of the opposite angles.

Sine Theorem

Evidence.

Let there be Δ ABC with sides a, b, c and angles α, β, γ.
Prove that

Sine Theorem

We will carry out from CD height CD. Then from Δ ACD we get:

Sine Theorem

If angle α is blunt, then

Sine Theorem

From Δ BCD we get

Sine Theorem

Similarly, we get

Sine Theorem

The theorem is proved.
created: 2014-10-05
updated: 2026-03-10
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Lectures and tutorial on "Planometry"

Terms: Planometry

Fundamentals of geometry
Perimeter
Planimetry
Straight and point
Section
Semi-straight beam.
Perpendicular straight lines
Parallel straight angles arising at the intersection.
A sign of parallel lines
Half plane
Angle
Types of angles
Angle bisector
Triangle
Types of triangles
Equality of triangles
The height, median and bisector of the triangle
The first sign of equality of triangles
The second sign of equality of triangles
Properties of parallel lines
Equilateral triangle
Properties of angles of an isosceles triangle
Sign of an isosceles triangle
Median property of an isosceles triangle
The property of the median and height of an isosceles triangle
The third sign of equality of triangles
The sum of the corners of the triangle
The outer corners of the triangle
Right triangle
The existence and uniqueness of the perpendicular to the line
Distance from point to straight
Circle
Touching Circles
Circle properties
Circumference described around a triangle
Middle perpendicular. Mediaatrix
Triangle Circumference
Building shapes
Construction of a triangle with given sides
Construction of an angle equal to this
Construction of the bisector of the angle
Dividing the segment in half
Construction of a perpendicular straight line
Construction of the fourth proportional segment
Quadrilateral
Parallelogram. Signs of parallelogram
Parallelogram properties
Rectangle. Rectangle Tag
Rectangle properties
Rhombus. Signs of rhombus.
Diamond property
Square. Sign of the square
Thales theorem
Center line of a triangle
Trapezium
Trapezoid properties
Proportional Sequence Theorem
Cosine angle
Pythagorean theorem
Triangle inequality
Sine and tangent angle
Cartesian coordinates on the plane
Distance between points
Mid section
Equation of circle
Equation of a straight line - analytical geometry
The coordinates of the point of intersection of straight lines
The position of the line relative to the coordinate system
The angular coefficient in the equation is straight. The geometric meaning of the coefficient.
Shape transformation
Motion properties
Symmetry about point
The symmetry property about a point
Symmetry relatively straight
Turn
Parallel transfer
Parallel Carry Property
Semi-direct directionality - theory and example
Similarity conversion
Vector
Unit vectors
Collinear Vector
Collinear vector. Properties
Vector coordinates
Coordinates of the vector. Properties
Equality of vectors
Addition of vectors. Parallelogram rule
Addition of vectors. Triangle rule
Addition of vectors. Difference
The property of multiplying a vector by a number
Multiply vector by number
Homothety
Homotetia. Property
Similarity of figures
Sign of the similarity of triangles at two angles
Sign of similarity of triangles on two sides and angles between them
Sign of similarity of triangles on three sides
Corners inscribed in a circle
Angles inscribed in a circle. Property
Proportionality of chord segments
Cosine theorem
Sine Theorem
Broken
Convex polygons
Convex polygons. Property
Regular polygons
Formula for the radii of inscribed circles of regular polygons
The formula for the radii of the circumscribed circles of regular polygons
Circumference
Radian angle measure
Square
Rectangle area
Parallelogram area
Area of ​​a triangle
Trapezium area
Circle, circular segment, circular sector
Area of ​​a circle
Area of ​​a circular sector
Circular segment area