$\frac{2r(sinA+sinB+sinC)}{4R} = 1-(sin^{2}\frac{B}{2}+sin^{2}\frac{C}{2}) + sin^{2}\frac{A}{2}$$\Leftrightarrow SINa +SINb + SINc = COSc + 2SIN^{2}\frac{a}{2}$
$\
Lef
tr
ighta
rrow $ 2
R(sinA
+
sinB
+
sinC)
/4R = 1-
[sin²(
B/2) + sin
²(C/2
)] + sin²(A/2)$\
Lef
tr
ighta
rrow $ sinA + sinB
+
sin
C = 2
sin²(A/2) + cosB + cosC$\
Lef
tr
ighta
rrow $ 2sinA/2.c
osA/2 + 2sin(B/2+C
/2
).cos(B/2-C/2)
= 2sin
²(A/2) + 2cos(B/2+C/2).cos(B/2
-C/2) $\
Lef
tr
ighta
rrow $ c
os(A/2).[cos(B/2+C/2) + cos(B/2-C/2)] = sin(A
/2).[cos(B/2+C/2) + cos(B/2
-C/2)] $\Leftrightarrow
$ 2.cos(A/2).cos(B/2).c
os(C/2) =
2sin(A/2).cos(B/2).cos(C
/2) (*) B, C là góc
tgiác nên 0 < B/2, C/2
< pi/2
=> cos(B/2), cos(C/2) > 0 (*) $\
Lef
tr
ighta
rrow $ c
os(A/2) = sin(A/2) => ta
n(A/2) = 1 => A/2 = $45^{0}
$ => A = $90^{
0}$
=> $\Delta $ABC vuông tại A