Liu Hui, a mathematician of Wei State in the Three Kingdoms period, put forward the method of "complementary entry and exit" to verify Pythagorean theorem when commenting on the ancient book Nine Chap

Liu Hui, a mathematician of Wei State in the Three Kingdoms period, put forward the method of "complementary entry and exit" to verify Pythagorean theorem when commenting on the ancient book Nine Chapters Arithmetic, as shown in the figure. Please say it. The side length of a square ABCD is A, and the point B is on AG.

The side length of the square EFGB is B, and the point C is on EB.

The side length of the square EHIA is c, and the point h is on FG.

Let IJ⊥AG cross j, HI cross AG cross k, AE cross CD cross l;

EA = EH = a，EB=EF=b，∠EBA=∠EFH=90，

∴ Rt△EFH≌Rt△EBA，∠ 1=∠2，FH=BA=a

∴ Rt△EFH，

Right angle FH=a, right angle EF=b, hypotenuse EH=c,

∠∠2 =∠3 =∠4 = 90-∠EAB，∠ 1=∠2，

∴ ∠ 1=∠3，EH=AI=a，∠ EFH =∠ AJI = 90，

∴ Rt△EFH≌Rt△AJI，JI=FH=a

∠∠5 =∠3 = 90-∠AIJ，∠3=∠4，

∴ ∠4=∠5, and DA=JI=a, ∠ ADL =∠ Ijk = 90,

∴ Rt△ADL≌Rt△IJK

∠∠6 =∠ 1 = 90-∠EHF，∠ 1=∠2，

∴ ∠2=∠6，EC=HB=b-a，∠ LCE =∠ KGH = 90。

∴rt△lce≌rt△kgh；

∴ To sum up: square ABCD area+square EFGB area

= square EHIA area;

Namely: a? +b？ =c？ ;

In a right triangle, the sum of the squares of two right-angled sides is equal to the square of the hypotenuse.