If a4+b4+c4 = 2c2(a2+b2), prove that C = 45o or 135o.
a4 + b4 +c4 = 2c2 (a2+b2)
or, (a2+b2)2 - 2a2b2 + c4 = 2(a2+b2)c2 [a2+ b2 = (a+b)2 - 2ab]
or, (a2+b2)2 - 2(a2+b2)c2 + (c2)2 = 2a2b2
or, (a2+ b2 - c2)2 = 2a2b2
or, a2+ b2 - c2= 2 ab
...
The Sine LawStatement: In any triangle the sides are proportional to the Sine of the opposite angles.In other words, in any ΔABC,
The Sine Law
Statement: In any triangle the sides are proportional to the Sine of the opposite angles.
In other words, in any ΔABC, 
Proof:
Let ABC be a triangle with a = BC, b = CA and c = AB, then three cases are possible. The angle C is either acute or right or obtuse angle.
Draw AD perpendicular to BC ( produce BC if necessary )
In ΔABC, SinB = 
for all figures
or, AD = cSinB → (i)
Also in ΔACD,
1 { SinC in fig (i), Sin90 = SinC in fig (ii) and Sin(π-C) = SinC in fig (iii) }
Therefore for all the figures,
SinC = 
AD = bSinC → (ii)
From (i) and (ii) we can write,
AD = bSinC = cSinB
↣ 
Similarly we can show,
Combining these we get,
Proved.
