∴mc=ac/2=3cm;
Similarly: CN=BC/2=2cm.
∴MN=MC+CN=3cm+2cm=5cm.
(2)MN=(a+b)/2。 (The calculation method is the same as above)
(3) The line segment length MN remains unchanged, which is proved as follows:
① When point C is on the AB line segment (see Figure ①):
If point m is the midpoint of AC, then MC = AC/2; Similarly: CN=BC/2.
∴mn=mc+cn=ac/2+bc/2=(ac+bc)/2=ab/2;
② When point C is on the extension line of line segment AB (see Figure ②):
If point m is the midpoint of AC, then MC = AC/2; Similarly: CN=BC/2.
∴mn=mc-cn=ac/2-bc/2=(ac-bc)/2=ab/2;
③ When point C is on the extension line of line segment BA (see Figure ③):
If point m is the midpoint of AC, then MC = AC/2; Similarly: CN=BC/2.
∴mn=cn-mc=bc/2-ac/2=(bc-ac)/2=ab/2.