The three sides of triangle $a < b < c$ are in arithmetic progression (AP) with common difference $d = b - a = c - b$. Denote the angles opposite to sides $a , b , c$ respectively by $A , B , C$.
Statements
(1) $d$ must be less than $a$.
(2) $d$ can be any positive number less than $a$.
(3) The numbers $\sin A , \sin B , \sin C$ are in AP.
(4) The numbers $\cos A , \cos B , \cos C$ are in AP.

12. In $\triangle A B C$, $a = 4 , b = 5 , c = 6$, then $\frac { \sin 2 A } { \sin C } =$ $\_\_\_\_$.

16. (This question is worth 14 points) In $\triangle ABC$, the sides opposite to angles $A , B , C$ are $a , b , c$ respectively. Given that $A = \frac { \pi } { 4 }$ and $b ^ { 2 } - a ^ { 2 } = \frac { 1 } { 2 } c ^ { 2 }$ . (I) Find the value of $\tan C$; (II) If the area of $\triangle ABC$ is 7, find the value of $b$.

In triangle $ABC$, the sides opposite to angles $A, B, C$ are $a, b, c$ respectively. Given that the area of $\triangle ABC$ is $\frac { a ^ { 2 } } { 3 \sin A }$.
(1) Find $\sin B \sin C$;
(2) If $b + c = 2$, find the range of values of $a$.

In $\triangle ABC$, the interior angles $A$, $B$, $C$ have opposite sides $a$, $b$, $c$ respectively. If $2b\cos B = a\cos C + c\cos A$, then $B = $ \_\_\_\_

In $\triangle A B C$, the sides opposite to angles $A$, $B$, $C$ are $a$, $b$, $c$ respectively. Given $b \sin C + c \sin B = 4 a \sin B \sin C$ and $b ^ { 2 } + c ^ { 2 } - a ^ { 2 } = 8$, then the area of $\triangle A B C$ is \_\_\_\_

In $\triangle ABC$, the sides opposite to angles $A, B, C$ are $a, b, c$ respectively. If the area of $\triangle ABC$ equals $\frac { a ^ { 2 } + b ^ { 2 } - c ^ { 2 } } { 4 }$, then $C =$
A. $\frac { \pi } { 2 }$
B. $\frac { \pi } { 3 }$
C. $\frac { \pi } { 4 }$
D. $\frac { \pi } { 6 }$

15. In $\triangle A B C$, the sides opposite to angles $A , B , C$ are $a , b , c$ respectively. Given $b \sin A + a \cos B = 0$, then $B =$ $\_\_\_\_$ .

18. (12 points) In $\triangle A B C$ , the sides opposite to angles $A , B , C$ are $a , b , c$ respectively. Given that $a \sin \frac { A + C } { 2 } = b \sin A$ .
(1) Find $B$ .
(2) If $\triangle A B C$ is an acute triangle and $c = 1$ , find the range of the area of $\triangle A B C$ .

In $\triangle A B C$ , $\sin ^ { 2 } A - \sin ^ { 2 } B - \sin ^ { 2 } C = \sin B \sin C$ .
(1) Find $A$ ;
(2) If $B C = 3$ , find the maximum value of the perimeter of $\triangle A B C$ .

Let the sides opposite to angles $A , B , C$ of $\triangle A B C$ be $a , b , c$ respectively. Given
$$\sin C \sin ( A - B ) = \sin B \sin ( C - A )$$
(1) If $A = 2 B$ , find $C$ ;
(2) Prove: $2 a ^ { 2 } = b ^ { 2 } + c ^ { 2 }$ .

(12 points) Let the sides opposite to angles $A, B, C$ of $\triangle ABC$ be $a, b, c$ respectively. Given $$\sin C \sin(A - B) = \sin B \sin(C - A)$$ (1) Prove: $2a^2 = b^2 + c^2$;
(2) If $a = 5, \cos A = \frac{25}{31}$, find the perimeter of $\triangle ABC$.

Let the interior angles $A , B , C$ of $\triangle A B C$ and their opposite sides $a , b , c$ satisfy $\sin A + \sqrt { 3 } \cos A = 2$.
(1) Find $A$.
(2) If $a = 2$ and $\sqrt { 2 } b \sin C = c \sin 2 B$, find the perimeter of $\triangle A B C$.

(13 points) Let the sides opposite to angles $A , B , C$ of $\triangle A B C$ be $a , b , c$ respectively. Given $\sin C = \sqrt { 2 } \cos B , a ^ { 2 } + b ^ { 2 } - c ^ { 2 } = \sqrt { 2 } a b$ .
(1) Find $B$ ;
(2) If the area of $\triangle A B C$ is $3 + \sqrt { 3 }$ , find $c$ .

Given that the area of $\triangle ABC$ is $\frac{1}{4}$, if $\cos 2A + \cos 2B + 2\sin C = 2$, $\cos A \cos B \sin C = \frac{1}{4}$, then
A. $\sin C = \sin^2 A + \sin^2 B$
B. $AB = \sqrt{2}$
C. $\sin A + \sin B = \frac{\sqrt{6}}{2}$
D. $AC^2 + BC^2 = 3$

Given that the area of $\triangle ABC$ is $\frac{1}{4}$, if $\cos 2A + \cos 2B + 2\sin C = 2$, $\cos A \cos B \sin C = \frac{1}{4}$, then
A. $\sin C = \sin^2 A + \sin^2 B$
B. $AB = \sqrt{2}$
C. $\sin A + \sin B = \frac{\sqrt{6}}{2}$
D. $AC^2 + BC^2 = 3$

In a right angle triangle with sides $a < b < c$, where $\angle ACB = \theta$ is the smallest angle, show that $\sin^2\theta - \sqrt{5}\sin\theta + 1 = 0$, given that $\frac{1}{a^2} + \frac{1}{b^2} = \frac{1}{c^2}$ (i.e., the reciprocals of the sides also form a right triangle).

If $A , B , C$ are the angles of a triangle and $\sin ^ { 2 } A + \sin ^ { 2 } B = \sin ^ { 2 } C$, then $C$ is equal to
(A) $30 ^ { \circ }$
(B) $90 ^ { \circ }$
(C) $45 ^ { \circ }$
(D) none of the above

If $A , B , C$ are the angles of a triangle and $\sin ^ { 2 } A + \sin ^ { 2 } B = \sin ^ { 2 } C$, then $C$ is equal to
(A) $30 ^ { \circ }$
(B) $90 ^ { \circ }$
(C) $45 ^ { \circ }$
(D) none of the above

If $A , B , C$ are the angles of a triangle and $\sin ^ { 2 } A + \sin ^ { 2 } B = \sin ^ { 2 } C$, then $C$ is equal to
(A) $30 ^ { \circ }$
(B) $90 ^ { \circ }$
(C) $45 ^ { \circ }$
(D) none of the above

In a triangle $ABC$, $3\sin A + 4\cos B = 6$ and $4\sin B + 3\cos A = 1$ hold. Then the angle $C$ equals
(A) $30^\circ$
(B) $60^\circ$
(C) $120^\circ$
(D) $150^\circ$.

In a triangle $A B C$ with fixed base $B C$, the vertex $A$ moves such that
$$\cos B + \cos C = 4 \sin ^ { 2 } \frac { A } { 2 }$$
If $a , b$ and $c$ denote the lengths of the sides of the triangle opposite to the angles $A , B$ and $C$, respectively, then
(A) $b + c = 4 a$
(B) $b + c = 2 a$
(C) locus of point $A$ is an ellipse
(D) locus of point $A$ is a pair of straight lines

If the angles $\mathrm { A } , \mathrm { B }$ and C of a triangle are in an arithmetic progression and if $\mathrm { a } , \mathrm { b }$ and c denote the lengths of the sides opposite to $\mathrm { A } , \mathrm { B }$ and C respectively, then the value of the expression $\frac { a } { c } \sin 2 C + \frac { c } { a } \sin 2 A$ is
A) $\frac { 1 } { 2 }$
B) $\frac { \sqrt { 3 } } { 2 }$
C) 1
D) $\sqrt { 3 }$

In a triangle the sum of two sides is $x$ and the product of the same two sides is $y$. If $x^2 - c^2 = y$, where $c$ is the third side of the triangle, then the ratio of the in-radius to the circum-radius of the triangle is
(A) $\frac{3y}{2x(x+c)}$
(B) $\frac{3y}{2c(x+c)}$
(C) $\frac{3y}{4x(x+c)}$
(D) $\frac{3y}{4c(x+c)}$

Column I
(A) In a triangle $\triangle X Y Z$, let $a , b$ and $c$ be the lengths of the sides opposite to the angles $X , Y$ and $Z$, respectively. If $2 \left( a ^ { 2 } - b ^ { 2 } \right) = c ^ { 2 }$ and $\lambda = \frac { \sin ( X - Y ) } { \sin Z }$, then possible values of $n$ for which $\cos ( n \pi \lambda ) = 0$ is (are)
(B) In a triangle $\triangle X Y Z$, let $a , b$ and $c$ be the lengths of the sides opposite to the angles $X , Y$ and $Z$, respectively. If $1 + \cos 2 X - 2 \cos 2 Y = 2 \sin X \sin Y$, then possible value(s) of $\frac { a } { b }$ is (are)
(C) In $\mathbb { R } ^ { 2 }$, let $\sqrt { 3 } \hat { i } + \hat { j } , \hat { i } + \sqrt { 3 } \hat { j }$ and $\beta \hat { i } + ( 1 - \beta ) \hat { j }$ be the position vectors of $X , Y$ and $Z$ with respect to the origin $O$, respectively. If the distance of $Z$ from the bisector of the acute angle of $\overrightarrow { O X }$ with $\overrightarrow { O Y }$ is $\frac { 3 } { \sqrt { 2 } }$, then possible value(s) of $| \beta |$ is (are)
(D) Suppose that $F ( \alpha )$ denotes the area of the region bounded by $x = 0 , x = 2 , y ^ { 2 } = 4 x$ and $y = | \alpha x - 1 | + | \alpha x - 2 | + \alpha x$, where $\alpha \in \{ 0,1 \}$. Then the value(s) of $F ( \alpha ) + \frac { 8 } { 3 } \sqrt { 2 }$, when $\alpha = 0$ and $\alpha = 1$, is (are) Column II (P) 1 (Q) 2 (R) 3 (S) 4 (T) 5