20 10-07-03
10:56
Option d
(1) Because ACD and BCE are isosceles right triangles, BC/BE=AC/AD.
So BC*AD=BE*AC
Because AD and CE are parallel, CN/AD=BC/AB, which means BC*AD=AB*CN.
In the same way, BE*AC=CM*AB.
To sum up AB*CN=CM*AB.
So CN=CM
Because MCN angle =90 degrees.
So NMC angle = NCBI angle =45 degrees.
So MN is parallel to AB.
(2) BEcause CD is parallel to Be.
So triangle CDN is similar to triangle BEN.
So NC/EN=CD/EB.
So NC/CD=EN/EB
And because CD=AD, EC=EB.
So NC/AD=EN/EC
Because CN and AD are parallel.
MN is parallel to AB
So NC/AD=BC/AB, EN/EC=MN/AC.
So BC/AB=MN/AC.
That is BC/(AC+BC)=MN/AC.
So1/Mn = (AC+BC)/AC * BC =1/AC+1/BC.
My supplement
20 10-07-03
1 1:33
(3) Let the midpoint of AB be O, so AO=BO.
According to (2),1/Mn =1/AC+1/BC.
So MN=AC*BC/(AC+BC)=AC*BC/AB.
To prove that Mn
Only need to prove AC * BC/AB.
Because AC=AO+OC, BC=OB-OC=AO-OC.
So you only need to prove (ao+oc) * (ao-oc)/ab.
That is, ao 2-oc 2 < =1/4ab 2.
1/4ab^2=( 1/2ab)^2=ao^2
OC 2 > =0
Clearly established
So the original proposition proves that