[Boutique] congruent triangle proof

Article 1: Congruent Triangular Proof

As shown in the figure, in △ABC and △ADE, AB=AC, AD=AE, ∠BAC=∠DAE=90°.

1 When point D is on the AC, as shown in Figure 1, what is the quantitative relationship and positional relationship between the line segments BD and CE? Write the conclusion of your guess directly; 2 Turn the △ADE in Figure 1 around the point A clockwise. Angle, as shown in Figure 2, what is the quantitative relationship and positional relationship between line segments BD and CE? Please explain why.

When △ABC and △ADE satisfy the following conditions of A, B, and C, the positional relationship between the line segments BD and CE is still established. It is not necessary to explain the reason.

A: AB: AC = AD: AE = 1, ∠ BAC = ∠ DAE ≠ 90 °; B: AB: AC = AD: AE ≠ 1, ∠ BAC = ∠ DAE = 90 °; C: AB: AC = AD: AE≠1, ∠BAC=∠DAE≠90°.

Part 2: Congruent Triangular Proof

When △DEC rotates around point C to the position shown in Figure 3, the relationship between S1 and S2 in Xiaoming's conjecture still holds, and attempts to make BC and CE on △BDC and △AEC respectively, please You prove Xiao Ming’s conjecture.

It is known that ∠ABC=60°, point D is a point on the angle bisector, BD=CD=4, DE∥AB is BC at point E. If there is a point F on the ray BA, make S△DCF=S△BDE, please Write the corresponding BF length directly.

Part 3: Congruent Triangular Proof

"Edge" axiom: three sides are equal to two equal triangles

"corner edge" axiom: two triangles equal to the angle between the two sides and its angle

"corner corner" axiom: two angles and their sides correspond to two equal triangles

The "corner edge" theorem: two triangles and the opposite sides of one of the corners are equal to the two triangles

"Corner-edge right-angled side" axiom: the oblique side and a right-angled edge correspond to two equal triangles

[Only right-angled triangles can be used with the right-angled sides of the beveled edges. There are three other types.] You can't use the right-angled sides of the beveled triangles. You can only use those three types.