Cosine theorem C 2 = A 2+B 2-2 ABCOSC

Substitute 2s = (a+b) 2-c 2.

Get absinC=2ab+2abcosC。

sinC = 2+2 OSC

Because (sinc) 2+(cosc) 2 = 1.

CosC=-3/5 sinC=4/5 tanC=-4/3。

Or cosC=- 1 sinC=0.

So tanC=-4/3

2. Solution: LGA-LGC = LG SINB =-LG radical 2

= & gtA/c=sinB= 1/ root number 2

= & gtB=45 degrees

Sine theorem sinA/sinC=a/c= 1/ root number 2.

= & gtSinC= radical number 2*sinA=sin(A+B)=(sinA+cosA)/ radical number 2.

=> Sina =cosA

= & gtA=45 degrees

So C= 180-A-B=90 degrees.

Therefore, a triangle is an isosceles right triangle.

3. Solution: By trigonometric function and sine and cosine theorem

sinC=sin(2A)=2sinAcosA

sinC/sinA=2cosA=3/2

a/sinA=c/sinC

c/a=sinC/sinA=3/2

c=3a/2

a+c= 10

a+3a/2= 10

a=4，c=6

b^2+c^2-a^2=2bccosA

b^2+36- 16=2b*6*3/4

b^2-9b+20=0

(b-4)(b-5)=0

B=4 or b=5.

But if b=4, then a=b=4, then the triangle is an isosceles triangle, and a = b = (1/2) c.

Then A+B+C=4A= 180, so a = b = 45 and c = 90. The triangle at this time is an isosceles right triangle.

But a 2+b 2 = 32, c 2 = 36, a 2+b 2 ≠ c 2,

From the inverse theorem of Pythagorean theorem, we know that a triangle is not a right triangle, which is contradictory!

So b=5(b=4 give up! )