Start the journey of mathematics by drawing on the great god's picture
First of all, let's start the math journey with the diagram of Cao Peiwei 00 1!
Find the vertical foot p
The problem is this: the position of E is a mystery, so we do NP⊥BE and find the vertical foot P.
Equal NP and BP
Because BN is the bisector of ∠CBE and ∠CBE is a right angle, ∠ PBN = 45. This means that NP and BP are equal!
Similar △ADM and △PMN
Then, DM⊥MN makes ∠ADM and ∠PMN equal. Through a series of reasoning, we prove that △ADM and △PMN are similar.
Magical conclusion
Even more amazing, because m is the midpoint of the square, we have 2AM=AD, and then we get 2PN=MP. Together with the previous conclusion, we get MB=BP=NP, so MP=DA!
Congruent triangles's proof
In similar triangles, if a set of corresponding edges are equal, then the two triangles are congruent! So △ ADM △ PMN, which means DM=MN. We proved it!
Practical thinking method
Finally, if m is any point, we can still get the same conclusion. This way of thinking is very practical. You can try to get congruent triangles with similar and equilateral sides. Come and have a try!