﻿ Proof the ratio of a side and the sines of its opposite angle of a triangle is equal to the diameter of the circumscribed circle of the triangle

# Proof the ratio of a side and the sines of its opposite angle of a triangle is equal to the diameter of the circumscribed circle of the triangle

The above statement is actually the law of sines, which tells us that the lengths of the sides of a triangle over the sines of their corresponding opposite angles are equal to a constant variable, i.,e., the diameter of the circumscribed circle of the triangle.

Considering three cases when the circumcenter is inside or outside the triangle, or on one side of the triangle, we will prove the equation, for any given side c and its opposite interior angle C in a triangle,

\dfrac{c}{\sin C} =2R ,

in which R is the radius of the circumcircle of the triangle.

## The circumcenter is inside the triangle

In this case, angle C and the central angle ∠AOB intersect the arc, so

∠C = \dfrac{1}{2} ∠AOB

As △AOB is an isosceles triangle, ∠AOD is the bisector of ∠AOB ,

therefore,

∠AOD = ∠C , \sin C = \sin∠AOD

\sin∠AOD= \dfrac{\dfrac{c}{2} }{AO} = \dfrac{c}{2R}

\therefore \dfrac{c}{\sin C} =2R

## The circumcenter is outside the triangle

In this case, ∠C is an obtuse angle, so the central angle intersecting the arc ∠AOB is an angle more than 180 \degree.

According to the Inscribed Angle Theorem,

∠C = \dfrac{1}{2} ∠AOB =∠AOG

\because ∠AOF = 180 \degree - ∠AOG

\therefore \sin ∠AOF = \sin ∠AOG= sin ∠C

\therefore sin ∠C = = \dfrac{\dfrac{c}{2} }{AO} = \dfrac{c}{2R}

\therefore \dfrac{c}{\sin C} =2R

## The circumcenter is on one side of the triangle

In this case,

∠C = 90\degree

and

c=2R

\dfrac{c}{\sin C} =\dfrac{2R}{\sin 90\degree } = 2R

## Summary

In summary, there are 3 cases based on the position of the circumcenter and the inscribed triangle,

1. the circumcenter is inside the triangle
2. the circumcenter is outside the triangle
3. the circumcenter is on one side of the triangle

The types of the triangle is also determined in he three cases , which are acute triangle, obtuse triangle and right triangle. In any of the cases, it's true that the ratio of any side and the sines of its opposite angle is equal to the diameter of the circumscribed circle.

Steven Zheng posted 3 years ago

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The text passage provides a detailed proof of the Law of Sines, which states that the ratio of a side of a triangle to the sine of its opposite angle is equal to the diameter of the circumscribed circle of the triangle. The proof is broken down into three cases based on the position of the circumcenter relative to the triangle.
no_question

the passage does not contain a mathematical question but rather provides a proof of a mathematical theorem. no specific question is asked, and no answer to a question is provided.
Given a text passage, determine:
    1. If it contains a mathematical question (including word problems or calculations)
    2. If an answer to that mathematical question is provided in the text

First, describe the content structure you found in 1-2 sentences.

Then provide your analysis in the following format:
TYPE: <one of the following values>
- NO_QUESTION
- SINGLE_QUESTION
- SINGLE_QUESTION_WITH_ANSWER
- MULTIPLE_QUESTIONS
- MULTIPLE_QUESTIONS_WITH_ANSWERS
system
﻿ Proof the ratio of a side and the sines of its opposite angle of a triangle is equal to the diameter of the circumscribed circle of the triangle

# Proof the ratio of a side and the sines of its opposite angle of a triangle is equal to the diameter of the circumscribed circle of the triangle

The above statement is actually the law of sines, which tells us that the lengths of the sides of a triangle over the sines of their corresponding opposite angles are equal to a constant variable, i.,e., the diameter of the circumscribed circle of the triangle.

Considering three cases when the circumcenter is inside or outside the triangle, or on one side of the triangle, we will prove the equation, for any given side c and its opposite interior angle C in a triangle,

\dfrac{c}{\sin C} =2R ,

in which R is the radius of the circumcircle of the triangle.

## The circumcenter is inside the triangle

In this case, angle C and the central angle ∠AOB intersect the arc, so

∠C = \dfrac{1}{2} ∠AOB

As △AOB is an isosceles triangle, ∠AOD is the bisector of ∠AOB ,

therefore,

∠AOD = ∠C , \sin C = \sin∠AOD

\sin∠AOD= \dfrac{\dfrac{c}{2} }{AO} = \dfrac{c}{2R}

\therefore \dfrac{c}{\sin C} =2R

## The circumcenter is outside the triangle

In this case, ∠C is an obtuse angle, so the central angle intersecting the arc ∠AOB is an angle more than 180 \degree.

According to the Inscribed Angle Theorem,

∠C = \dfrac{1}{2} ∠AOB =∠AOG

\because ∠AOF = 180 \degree - ∠AOG

\therefore \sin ∠AOF = \sin ∠AOG= sin ∠C

\therefore sin ∠C = = \dfrac{\dfrac{c}{2} }{AO} = \dfrac{c}{2R}

\therefore \dfrac{c}{\sin C} =2R

## The circumcenter is on one side of the triangle

In this case,

∠C = 90\degree

and

c=2R

\dfrac{c}{\sin C} =\dfrac{2R}{\sin 90\degree } = 2R

## Summary

In summary, there are 3 cases based on the position of the circumcenter and the inscribed triangle,

1. the circumcenter is inside the triangle
2. the circumcenter is outside the triangle
3. the circumcenter is on one side of the triangle

The types of the triangle is also determined in he three cases , which are acute triangle, obtuse triangle and right triangle. In any of the cases, it's true that the ratio of any side and the sines of its opposite angle is equal to the diameter of the circumscribed circle.

Steven Zheng posted 3 years ago

Scroll to Top
user
The text passage provides a detailed proof of the Law of Sines, which states that the ratio of a side of a triangle to the sine of its opposite angle is equal to the diameter of the circumscribed circle of the triangle. The proof is broken down into three cases based on the position of the circumcenter relative to the triangle.

TYPE: NO_QUESTION

The passage does not contain a mathematical question but rather provides a proof of a mathematical theorem. No specific question is asked, and no answer to a question is provided.