Ta có:
S Δ A M N = 1 2 d ( N , S A ) . A M S_{ΔAMN}=\dfrac{1}{2}d{(N,SA)}.AM S Δ A M N = 2 1 d ( N , S A ) . A M S Δ S M N = 1 2 d ( N , S A ) . S M S_{ΔSMN}=\dfrac{1}{2}d{(N,SA)}.SM S Δ S M N = 2 1 d ( N , S A ) . S M A M = S M AM=SM A M = S M
⇒ S Δ A M N = S Δ S M N \Rightarrow S_{ΔAMN}=S_{ΔSMN} ⇒ S Δ A M N = S Δ S M N
Ta lại có: S M S A = S N S B = 1 2 \dfrac{SM}{SA}=\dfrac{SN}{SB} =\dfrac{1}{2} S A S M = S B S N = 2 1
⇒ S Δ S M N S Δ S A B = 1 2 S M . S N . sin M S N ^ 1 2 S A . S B . sin A S B ^ = 1 4 ⇒ \dfrac{S_{ΔSMN}}{ S_{ΔSAB}}=\dfrac{\frac{1}{2}SM.SN.\sin\widehat{MSN}}{\frac{1}{2}SA.SB.\sin\widehat{ASB}}=\dfrac{1}{4} ⇒ S Δ S A B S Δ S M N = 2 1 S A . S B . sin A S B 2 1 S M . S N . sin M S N = 4 1
Do C D / / A B ⊂ ( S A B ) ⇒ C D / / ( S A B ) ⇒ d ( P ; ( S A B ) ) = d ( D ; ( S A B ) ) CD//AB⊂(SAB) ⇒CD//(SAB)⇒d(P;(SAB))=d(D;(SAB)) C D / / A B ⊂ ( S A B ) ⇒ C D / / ( S A B ) ⇒ d ( P ; ( S A B ) ) = d ( D ; ( S A B ) )
⇒ V A M N P = 1 3 . d ( P ; ( A M N ) ) . S Δ A M N = 1 3 . d ( D ; ( S A B ) ) . 1 4 S Δ S A B \Rightarrow V_{AMNP}=\dfrac{1}{3}. d(P;(AMN)).S_{ΔAMN}=\dfrac{1}{3}.d(D;(SAB)).\dfrac{1}{4} S_{ΔSAB} ⇒ V A M N P = 3 1 . d ( P ; ( A M N ) ) . S Δ A M N = 3 1 . d ( D ; ( S A B ) ) . 4 1 S Δ S A B
= 1 4 . V D S A B = 1 4 V S A B D = 1 4 . 1 2 . V = 1 8 V =\dfrac{1}{4} .V_{DSAB}=\dfrac{1}{4}V_{SABD}=\dfrac{1}{4}.\dfrac{1}{2}.V=\dfrac{1}{8}V = 4 1 . V D S A B = 4 1 V S A B D = 4 1 . 2 1 . V = 8 1 V
