The maximum problem in geometry refers to solving the minimum or maximum value of the sum of line segments or distances in a given geometry.

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This kind of questions can not only examine students' mastery of geometric properties and theorems, but also examine students' familiarity with graphics.

The common methods to solve the geometric maximum problem are:

1. Apply the axiom of the shortest line segment between two points (including applying the trilateral relationship of a triangle) to find the maximum value: In this case, we can use the properties of a triangle, such as the sum of two sides of a triangle is greater than the third side, to find the maximum value.

2. Apply the shortest property of vertical line segment to find the maximum value: In this case, we can use the distance formula from point to straight line and the property that straight line is perpendicular to straight line to find the maximum value.

3. Apply the property of axial symmetry to find the maximum value: the property of axial symmetry can help us find the symmetrical point in the graph, thus simplifying the problem and solving the maximum value.

4. Apply quadratic function to get the maximum value: establish coordinate system, transform geometric problem into quadratic function problem, and get the maximum value of geometric figure by solving the maximum value of quadratic function.

5. Apply other knowledge to find the maximum: for example, the maximum problem in analytic geometry can be solved by using knowledge such as inequality and trigonometric function.

Here is a concrete example:

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As shown in the figure, connect CP and find the minimum value of AC+PE when ABC rotates to A 1B2C position.

Problem solving process:

1. Establish a coordinate system, and let A(x 1, y 1), B(x2, y2), C(x3, y3), P(x0, y0).

2. According to the meaning of the question, find the coordinates of A 1, B2, C 1.

3. Find the minimum value of AC+PE with quadratic function. Let f(x)=AC+PE and find the minimum value of f(x).

4. The minimum value of f (x) is obtained by derivation and simplification.

5. It is concluded that the minimum value of AC+PE is the minimum value of f(x).

Through the above steps, we can solve the maximum problem in geometry. In practical application, according to the subject conditions and known knowledge, the appropriate method is flexibly selected to solve the maximum value.