오른쪽 그림과 같이 삼각형 A B C \mathrm{ABC} ABC A B \mathrm{AB} AB 2 : 1 2:1 2 : 1 P \mathrm P P A C \mathrm{AC} AC Q \mathrm Q Q R \mathrm R R A P Q \mathrm{APQ} APQ P R Q \mathrm{PRQ} PRQ P B C R \mathrm{PBCR} PBCR a a a d d d C Q ‾ / A R ‾ \overline{\mathrm{CQ}}/\overline{\mathrm{AR}} CQ / AR a a a d d d
삼각형 A P Q \mathrm{APQ} APQ a a a A P R \mathrm{APR} APR 2 a + d 2a+d 2 a + d
a : 2 a + d = △ A P Q : △ A P R a:2a+d=\triangle\mathrm{APQ}:\triangle\mathrm{APR} a : 2 a + d = △ APQ : △ APR = 1 2 A P ‾ ⋅ A Q ‾ sin A : 1 2 A P ‾ ⋅ A R ‾ sin A =\frac12\overline{\mathrm{AP}}\cdot\overline{\mathrm{AQ}}\sin A:\frac12\overline{\mathrm{AP}}\cdot\overline{\mathrm{AR}}\sin A = 2 1 AP ⋅ AQ sin A : 2 1 AP ⋅ AR sin A 가 성립한다. 따라서
A Q ‾ A R ‾ = a 2 a + d ⋯ ㉠ \frac{\overline{\mathrm{AQ}}}{\overline{\mathrm{AR}}}=\frac a{2a+d}\quad\cdots\text{㉠} AR AQ = 2 a + d a ⋯ ㉠ 같은 방법으로, 삼각형 A B C \mathrm{ABC} ABC (가) \boxed{\text{(가)}} ( 가 )
a : (가) = △ A P Q : △ A B C a:\boxed{\text{(가)}}=\triangle\mathrm{APQ}:\triangle\mathrm{ABC} a : ( 가 ) = △ APQ : △ ABC = 1 2 A P ‾ ⋅ A Q ‾ sin A : 1 2 A B ‾ ⋅ A C ‾ sin A =\frac12\overline{\mathrm{AP}}\cdot\overline{\mathrm{AQ}}\sin A:\frac12\overline{\mathrm{AB}}\cdot\overline{\mathrm{AC}}\sin A = 2 1 AP ⋅ AQ sin A : 2 1 AB ⋅ AC sin A 또한 점 P \mathrm P P A B \mathrm{AB} AB 2 : 1 2:1 2 : 1
A P ‾ = 2 3 A B ‾ \overline{\mathrm{AP}}=\frac23\overline{\mathrm{AB}} AP = 3 2 AB 따라서
A Q ‾ A C ‾ = (나) \frac{\overline{\mathrm{AQ}}}{\overline{\mathrm{AC}}}=\boxed{\text{(나)}} AC AQ = ( 나 ) 그러므로
C Q ‾ A Q ‾ = A C ‾ − A Q ‾ A Q ‾ = (다) ⋯ ㉡ \frac{\overline{\mathrm{CQ}}}{\overline{\mathrm{AQ}}}=\frac{\overline{\mathrm{AC}}-\overline{\mathrm{AQ}}}{\overline{\mathrm{AQ}}}=\boxed{\text{(다)}}\quad\cdots\text{㉡} AQ CQ = AQ AC − AQ = ( 다 ) ⋯ ㉡ ㉠, ㉡에 의해
C Q ‾ A R ‾ = C Q ‾ A Q ‾ ⋅ A Q ‾ A R ‾ = a + 2 d 2 a + d \frac{\overline{\mathrm{CQ}}}{\overline{\mathrm{AR}}}=\frac{\overline{\mathrm{CQ}}}{\overline{\mathrm{AQ}}}\cdot\frac{\overline{\mathrm{AQ}}}{\overline{\mathrm{AR}}}=\frac{a+2d}{2a+d} AR CQ = AQ CQ ⋅ AR AQ = 2 a + d a + 2 d 위의 과정에서 (가), (나), (다)에 알맞은 것은? [4점]
(가) (나) (다) \text{(가)}\qquad\text{(나)}\qquad\text{(다)} ( 가 ) ( 나 ) ( 다 )
참고 그림 · 출처: https://t1.daumcdn.net/cfile/tistory/994887345AE2500222
