how can you show that diameter of a circumscribe circle is equal a/sin(a)?

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sciencesolve | Teacher | (Level 3) Educator Emeritus

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You should draw a triangle ABC inscribed in a circle whose radius is r. You need to draw a diameter that starts from the vertex A and it ends in the point D.

Joining the points A, B, D, you'll get a right triangle since hat B is opposed to diameter AD.

You need to express the sine of the angle `hat(ADB) `  such that:

`sin hat(ADB) = (AB)/(AD)`

Notice that `AD`  represents the diameter of circle, hence `AD = d = 2r.`

Notice that the inscribed angles `hat(ACB) ` and `hat(ADB)`  cut the same chord, hence `hat(ACB) = hat(ADB).`

Since `sin hat(ADB) = (AB)/(AD) => sinhat(ACB) = (AB)/(AD) = (AB)/(d) => sin hatC = (AB)/(d) `

You may come up with the following substitutions for the sides of triangle ABC such that:

`AB = c`  (opposed to `hatC` )

`AC = b`  (opposed to `hatB` )

`BC = a (opposed hatA)`

Hence, usiing these new notations yields:

`sin hatC = c/(d) => c/(sin hat C) = d`

Resoning by analogy yields:

`c/(sin hat C) = a/(sin hat A) = b/(sin hat B) =d`

Hence, using the sine formula and circle identities you may prove the law of sines such that: `c/(sin hat C) = a/(sin hat A) = b/(sin hat B) = d.`

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