How to compute the values of angles A, B, and C all at once while keeping the equation unchanged, and efficiently obtain the results? A BUG?

Question

Clear["`*"]
Reduce[{Sin[C] == Sqrt[2] Cos[B], a^2 + b^2 - c^2 == Sqrt[2] a  b, 
  A + B + C == \[Pi], 0 < B < \[Pi], a/Sin[A] == b/Sin[B] == c/Sin[C],
   a^2 == b^2 + c^2 - 2 b  c  Cos[A], 
  b^2 == a^2 + c^2 - 2 a  c  Cos[B], 
  c^2 == b^2 + a^2 - 2  a  b  Cos[C], {a, b, c} > 0}, {A, B, C}, {a, 
  b, c}]

The software kept showing "running" without producing a result when solving this equation. However, the values of these three angles can be computed manually.

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An alternative approach can be used to find the values of these three angles.

Clear["`*"]
Reduce[{Sin[C] == Sqrt[2] Cos[B], Cos[C] == 1/Sqrt[2], 
    A + B + C == \[Pi], 0 < {A, B, C} < \[Pi]}, {A, B, C}] // 
  FullSimplify // Reduce

Manually substituting a known condition involving the relationship of sides using the cosine rule to obtain the cosine value of angle C.

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But my question is, how can the software be enabled to quickly produce results while keeping the equation in the first code unchanged?

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The strange thing is that, while keeping the equation unchanged, the software can solve for each of the three angles individually every time.

 Clear["`*"]
Reduce[{Sin[C] == Sqrt[2] Cos[B], a^2 + b^2 - c^2 == Sqrt[2] a  b, 
    A + B + C == \[Pi], 0 < {A, B, C} < \[Pi], 
    a/Sin[A] == b/Sin[B] == c/Sin[C], 
    a^2 == b^2 + c^2 - 2 b  c  Cos[A], 
    b^2 == a^2 + c^2 - 2 a  c  Cos[B], 
    c^2 == b^2 + a^2 - 2  a  b  Cos[C], {a, b, c} > 0}, {A}, {a, b, c,
     B, C}] // FullSimplify // Reduce

A == (5 \[Pi])/12

 Clear["`*"]
Reduce[{Sin[C] == Sqrt[2] Cos[B], a^2 + b^2 - c^2 == Sqrt[2] a  b, 
    A + B + C == \[Pi], 0 < {A, B, C} < \[Pi], 
    a/Sin[A] == b/Sin[B] == c/Sin[C], 
    a^2 == b^2 + c^2 - 2 b  c  Cos[A], 
    b^2 == a^2 + c^2 - 2 a  c  Cos[B], 
    c^2 == b^2 + a^2 - 2  a  b  Cos[C], {a, b, c} > 0}, {B}, {a, b, c,
     A, C}] // FullSimplify // Reduce

B == \[Pi]/3

Clear["`*"]
Reduce[{Sin[C] == Sqrt[2] Cos[B], a^2 + b^2 - c^2 == Sqrt[2] a  b, 
    A + B + C == \[Pi], 0 < {A, B, C} < \[Pi], 
    a/Sin[A] == b/Sin[B] == c/Sin[C], 
    a^2 == b^2 + c^2 - 2 b  c  Cos[A], 
    b^2 == a^2 + c^2 - 2 a  c  Cos[B], 
    c^2 == b^2 + a^2 - 2  a  b  Cos[C], {a, b, c} > 0}, {C}, {a, b, c,
     A, B}] // FullSimplify // Reduce

C == \[Pi]/4

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However, when attempting to compute the values of all three angles A, B, and C simultaneously, the software keeps displaying "running" without producing a result.

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Unable to decide whether numeric quantity -Pi + <<2>> - I (Log[1 - I
(Divide[-3 Power[2, <<1>>], (1 - Power[Root[{<<2>>}, {2, 3}], 2])
<<2>>] + <<124>>)] - Log[1 + <<1>>]) is equal to zero. Assuming it is.

Answer 1

Solve[{Sin[C] == Sqrt[2]  Cos[B], a^2 + b^2 - c^2 == Sqrt[2]  a  b, 
    A + B + C == \[Pi], 0 < {A, B, C} < \[Pi], 
    a/Sin[A] == b/Sin[B] == c/Sin[C], 
    a^2 == b^2 + c^2 - 2  b  c  Cos[A], 
    b^2 == a^2 + c^2 - 2  a  c  Cos[B], 
    c^2 == a^2 + b^2 - 2  a  b  Cos[C], {a, b, c} > 0}, {A, B, C, a, 
    b, c}] // FullSimplify // Simplify[#, a > 0] &

Solve[{Sin[C] == Sqrt[2]  Cos[B], a^2 + b^2 - c^2 == Sqrt[2]  a  b, 
    A + B + C == \[Pi], 0 < {A, B, C} < \[Pi], 
    a/Sin[A] == b/Sin[B] == c/Sin[C], 
    a^2 == b^2 + c^2 - 2  b  c  Cos[A], 
    b^2 == a^2 + c^2 - 2  a  c  Cos[B], 
    c^2 == a^2 + b^2 - 2  a  b  Cos[C], {a, b, c} > 0}, {A, B, C, a, 
    b, c}] // FullSimplify // Assuming[a > 0, Refine[#]] &