As shown, any triangle ABC is the circumscribed circle O of ABC.

Make the diameter BD cross ⊙O in d.

Connect DA.

∠DAB=90 degrees, because the circumferential angle of the inner diameter of the same circle or equal circle is a right angle.

Because the circumferential angles of the same arc in the same circle or equal circle are equal, ∠D is equal to ∠ACB.

So c/sinC=c/sinD=BD=2R.

Similar to the other two equations.