如圖所示,△ABC中,|AB|=2,|BC|=3.3個(gè)紅色三角形面積之和即為所求.
S△ADE=(1/2)|AE|*|AD|*sin∠DAE=|AC|sin∠CAB
S△BFG=(1/2)|BF|*|BG|*sin∠FBG=3sin∠ABC
S△CHI=(1/2)|CH|*|CI|*sin∠HCI=(3/2)|AC|sin∠BCA
在△ABC中,由正弦定理有：|AC|/sin∠ABC=3/sin∠CAB=2/sin∠BCA
故3個(gè)三角形面積之和=|AC|sin∠CAB+3sin∠ABC+(3/2)|AC|sin∠BCA=3*3sin∠ABC≤9*1=9
當(dāng)且僅當(dāng)∠ABC=90°時(shí)取“=”.
