Cn mathematical problems

(1) solution: ∫ point m is the midpoint of AC.

∴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.