P ( − sin ( β − α ) , − cos β ) P\bigl(-\sin(\beta-\alpha),\ -\cos\beta\bigr) P ( − sin ( β − α ) , − cos β ) Q ( cos ( β − α ) , sin β ) , R ( cos ( β − α + θ ) , sin ( β − θ ) ) Q\bigl(\cos(\beta-\alpha),\ \sin\beta\bigr),\quad R\bigl(\cos(\beta-\alpha+\theta),\ \sin(\beta-\theta)\bigr) Q ( cos ( β − α ) , sin β ) , R ( cos ( β − α + θ ) , sin ( β − θ ) ) 0 < α , β , θ < π 4 0 < \alpha,\ \beta,\ \theta < \dfrac{\pi}{4} 0 < α , β , θ < 4 π To find: which point lies on the segment, or non-collinear? \text{To find: which point lies on the segment, or non-collinear?} To find: which point lies on the segment, or non-collinear? Plan: collinear ⇔ area ( △ P Q R ) = 0 \text{Plan: collinear} \Leftrightarrow \text{area}(\triangle PQR) = 0 Plan: collinear ⇔ area ( △ P QR ) = 0 Δ = ( x Q − x P ) ( y R − y P ) − ( y Q − y P ) ( x R − x P ) \Delta = (x_Q-x_P)(y_R-y_P) - (y_Q-y_P)(x_R-x_P) Δ = ( x Q − x P ) ( y R − y P ) − ( y Q − y P ) ( x R − x P ) x Q − x P = cos ( β − α ) + sin ( β − α ) x_Q-x_P = \cos(\beta-\alpha) + \sin(\beta-\alpha) x Q − x P = cos ( β − α ) + sin ( β − α ) y Q − y P = sin β + cos β y_Q-y_P = \sin\beta + \cos\beta y Q − y P = sin β + cos β x R − x P = cos ( β − α + θ ) + sin ( β − α ) , y R − y P = sin ( β − θ ) + cos β x_R-x_P = \cos(\beta-\alpha+\theta) + \sin(\beta-\alpha),\quad y_R-y_P = \sin(\beta-\theta) + \cos\beta x R − x P = cos ( β − α + θ ) + sin ( β − α ) , y R − y P = sin ( β − θ ) + cos β p = β − α , q = β p = \beta-\alpha,\quad q = \beta p = β − α , q = β Δ = ( cos p + sin p ) ( sin ( q − θ ) + cos q ) − ( sin q + cos q ) ( cos ( p + θ ) + sin p ) \Delta = (\cos p+\sin p)(\sin(q-\theta)+\cos q) - (\sin q+\cos q)(\cos(p+\theta)+\sin p) Δ = ( cos p + sin p ) ( sin ( q − θ ) + cos q ) − ( sin q + cos q ) ( cos ( p + θ ) + sin p ) = ( 1 − sin θ − cos θ ) ( cos p cos q − sin p sin q ) = (1-\sin\theta-\cos\theta)(\cos p\cos q - \sin p\sin q) = ( 1 − sin θ − cos θ ) ( cos p cos q − sin p sin q ) cos p cos q − sin p sin q = cos ( p + q ) = cos ( 2 β − α ) \cos p\cos q - \sin p\sin q = \cos(p+q) = \cos(2\beta-\alpha) cos p cos q − sin p sin q = cos ( p + q ) = cos ( 2 β − α ) Δ = cos ( 2 β − α ) ( 1 − sin θ − cos θ ) \Delta = \cos(2\beta-\alpha)\,(1-\sin\theta-\cos\theta) Δ = cos ( 2 β − α ) ( 1 − sin θ − cos θ ) 2 β − α ∈ ( − π 4 , π 2 ) ⇒ cos ( 2 β − α ) > 0 2\beta-\alpha \in \left(-\tfrac{\pi}{4},\ \tfrac{\pi}{2}\right) \Rightarrow \cos(2\beta-\alpha) > 0 2 β − α ∈ ( − 4 π , 2 π ) ⇒ cos ( 2 β − α ) > 0 θ ∈ ( 0 , π 4 ) ⇒ 1 < sin θ + cos θ < 2 ⇒ 1 − sin θ − cos θ < 0 \theta \in \left(0,\ \tfrac{\pi}{4}\right) \Rightarrow 1 < \sin\theta+\cos\theta < \sqrt{2} \Rightarrow 1-\sin\theta-\cos\theta < 0 θ ∈ ( 0 , 4 π ) ⇒ 1 < sin θ + cos θ < 2 ⇒ 1 − sin θ − cos θ < 0 Δ < 0 ⇒ area ≠ 0 ⇒ P , Q , R are non-collinear \Delta < 0 \Rightarrow \text{area} \neq 0 \Rightarrow P, Q, R \text{ are non-collinear} Δ < 0 ⇒ area = 0 ⇒ P , Q , R are non-collinear option (4) \boxed{\text{option (4)}} option (4)
