It is known that ∠ ace = ∠ CDE = 90 °, point B is on CE, CA = CB = CD, through the circle intersection ab of a, C and D to f (as shown in the figure). It is proved that f is the heart of △ CDE

It is known that ∠ ace = ∠ CDE = 90 °, point B is on CE, CA = CB = CD, through the circle intersection ab of a, C and D to f (as shown in the figure). It is proved that f is the heart of △ CDE

Proof: proof 1: as shown in the figure, if DF is connected, then FB = FD is obtained from the known, ∵ CDF = ∵ cab = 45 ° = 12 ∵ CDE, ∵ DF is the bisector of ∵ CDE, connecting BD and cf. if CD = CB, then ∵ FBD = ∵ cbd-45 ° = ∵ cdb-45 ° = ∵ FDB, then FB = FD is obtained, that is, the distance from F to B and D is equal, f is on the vertical bisector of line BD, and thus on the top bisector of isosceles triangle CBD, CF is the bisector of ∵ ECD, and ∵ f is △ CD The intersection of two angular bisectors on e is the heart of △ CDE. Proof 2: the same as proof 1, it is concluded that after ∠ CDF = 45 ° = 90 ° - 45 ° = ∠ FDE, there are four points of B, e, D and F in common circle because ∠ ABC = ∠ FDE. Connecting EF, after ∠ FBD = ∠ FDB is proved, there is ∠ fed = ∠ FBD = ∠ FDB = ∠ Feb immediately, that is, EF is the bisector of ∠ CED. Originally, there is little information about point E, so it should be proved that EF is angular bisector From this proof, we can see that f is the outer center of △ DCB. ∠ CDF = ∠ cab = 45 ° = 12 ∠ CDE, we know that DF is the bisector of ∠ CDE, so f is the inner part of △ CDE. Proof 3: as shown in the figure, only CF is the bisector of ∠ DCE= Proof 4: first, DF is the bisector of ∠ CDE, so the outer center I of △ CDE is on the straight line DF. Now, the coordinate system is established with Ca as the Y axis and CB as the X axis, and Ca = CB = CD = D, then the straight line AB is the image of the first-order function y = - x + D ① (as shown in the figure). If the coordinates of inner I are (x1, Y1), then x1+ Y1 = ch + IH = ch + HB = CB = D satisfies (1), that is, I is on the straight line AB, but I is on DF, so I is the intersection of AB and DF. From the uniqueness of the intersection, we know that I is f, so we can prove that F is the heart of RT △ CDE. We can also extend ed intersection ⊙ o at P1, and CP is the diameter