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2018 JHMT
6
2018 JHMT Geometry #6
2018 JHMT Geometry #6
Source:
September 5, 2023
geometry
Problem Statement
△
A
B
C
\vartriangle ABC
△
A
BC
is inscribed in a unit circle. The three angle bisectors of
A
A
A
,
B
B
B
,
C
C
C
are extended to intersect the circle at
A
1
A_1
A
1
,
B
1
B_1
B
1
,
C
1
C_1
C
1
, respectively. Find
A
A
1
cos
A
2
+
B
B
1
cos
B
2
+
C
C
1
cos
C
2
sin
A
+
sin
B
+
sin
C
.
\frac{AA_1 \cos \frac{A}{2} + BB_1 \cos \frac{B}{2} + CC_1 \cos \frac{C}{2}}{\sin A + \sin B + \sin C}.
sin
A
+
sin
B
+
sin
C
A
A
1
cos
2
A
+
B
B
1
cos
2
B
+
C
C
1
cos
2
C
.
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