After thinking, Xiao Ming showed a correct idea: as shown in Figure 2, extend CB to E, make BE=CD, connect AE, and prove ABEADC, so it is easy to prove that ACE is an equilateral triangle, so AC=CE, so AC = BC+CD.
Xiao Liang showed another correct way of thinking: as shown in Figure 3, rotate ABC 60 counterclockwise around point A to make AB and AD coincide, so it is easy to prove that ACF is an equilateral triangle, so AC=CF, so AC = BC+CD.
On this basis, the students did further research:
(1) Xiaoying proposed: As shown in Figure 4, if "ACB=ACD=ABD=ADB=60" is changed to "ACB=ACD=ABD=ADB=45", other conditions being unchanged, what is the equivalent relationship among line segments BC, CD and AC? Please write a conclusion about Xiaoying's question and give proof.
(2) Xiaohua proposed: As shown in Figure 5, if "ACB=ACD=ABD=ADB=60" is changed to "ACB=ACD=ABD=ADB=", and other conditions remain unchanged, what is the equivalent relationship among line segments BC, CD and AC? Please write an unproven conclusion about Xiaohua's problem.
Stem analysis:
(1) First judge ADE=ABC, you can get that ACE is an isosceles triangle, and then AEC=45, you can get an isosceles right triangle. (judging ADE=ABC, you can also judge the four-point circle of points A, B, C and D first. )
(2) Judge ADE=ABC first, then you can conclude that ACE is an isosceles triangle, and then use trigonometric function to draw a conclusion.
Thinking about solving problems:
This topic, entitled Geometric Transformation Synthesis, mainly examines the judgment of congruent triangles, the sum of internal angles of quadrilateral and isosceles triangle and their properties. The key to solve this problem is to construct congruent triangles, which is a basic problem.