如图,等边三角形ABC外接圆半径OC=R,内接圆半径OD=r,△ABC的边长为a,求r :a:R

如图,等边三角形ABC外接圆半径OC=R,内接圆半径OD=r,△ABC的边长为a,求r :a:R
如图,等边三角形ABC外接圆半径OC=R,内接圆半径OD=r,△ABC的边长为a,求r :a:R

如图,等边三角形ABC外接圆半径OC=R,内接圆半径OD=r,△ABC的边长为a,求r :a:R
设D在AC上,
OD垂直于AC,∠ACB=60°,∠OCD=30°,
OC=2OD
R=2r,
CD=AC/2=a/2,
OC²-OD²=CD²
4r²-r²=a²/4,
a=（2√3）r,
r:a:R=r:（2√3）r:2r=1:2√3:2.

r :a:R=√3／2×1／3a：a:√3／2a×2／3=1:2√3:2