A, isosceles (edge) triangle problem:

Typical example: The copyright belongs to Jinyuan Mathematics Studio and cannot be reproduced.

Example 1: (20 12, Chongzuo, Guangxi, 10) As shown in the figure, the vertex coordinates of parabola (a≠0) are points (-2,3), and the parabola intersects the Y axis at point B (0 0,2). (1) Find the parabola. (2) Whether there is a point P on the X axis that makes △PAB an isosceles triangle, and if there is, request the coordinates of the point P; If it does not exist, please explain the reason;

(3) If point P is any point on the X axis, find the coordinates of point P when PA-Pb is maximum.

Example 2: (20 12, Chaoyang, Liaoning, 14) is known. As shown in the figure, in the plane rectangular coordinate system, the hypotenuse BC of Rt△ABC is on the X axis, the right-angle vertex A is on the positive semi-axis of the Y axis, and A (0 0,2) and B (- 1 0).

(1) Find the coordinates of point C; (2) Find the analytical formula and symmetry axis of parabola passing through points A, B and C;

(3) Let point P(m, n) be the point of parabola in the first quadrant, and the area of △PAC be S. Find the functional relationship between S and M, and find the coordinates of point P with the largest S;

(4) Is there such a point M on the parabola axis of symmetry that △MPC(P is the point where S is the largest in the above question (3)) is an isosceles triangle? If it exists, please write the coordinates of point M directly; If it does not exist, please explain why.

Example 3: (20 12 Linyi, Shandong 13) As shown in the figure, point A is on the X axis, and OA=4. Rotate the line segment OA 120 clockwise around the o point to the OB position. (1) Find the coordinates of point B; (2) Find the analytical formula of parabola passing through points A, O and B;

(3) Is there a point P on the axis of symmetry of this parabola, so that the triangle with points P, O and B as its vertices is an isosceles triangle? If it exists, find the coordinates of point P; If it does not exist, explain why.

Example 4: (20 12, Baotou, Inner Mongolia, 12) It is known that the straight line y = 2x+4 intersects the X axis and the Y axis at points A and D respectively, the parabola passes through points A and D, and point B is another intersection point of the parabola and the X axis.

(1) Find the analytical expression of this parabola and the coordinates of point B;

(2) that set point m is a point on the straight line AD, and the coordinate of the point m is found;

(3) If point C(2, y) is on this parabola, is there a point P on the positive semi-axis of Y axis, so that △BCP is an isosceles triangle? If it exists, request the coordinates of point P; If it does not exist, please explain why.

Example 5: (20 12, Longyan, Fujian, 14) In the plane rectangular coordinate system xoy, as shown in the figure, a triangular plate with a 60-degree angle is placed, with the hypotenuse AB on the X axis and the right-angle vertex C on the positive semi-axis of the Y axis, and the known point A (- 1 0).

(1) Please write the coordinates of point B and point C directly: b (,), c (,); And find the parabolic analytical formula of a, b and c points;

(2) The existing triangle DEF (where ∠ edF = 90, ∠ DEF = 60) is exactly the same as the above triangle, and the vertex E is placed on the line segment AB (point E is a moving point that is not coincident with points A and B), and the straight line where ED is located passes through point C, and the straight line where EF is located is equal to (1).

(1) let AE=x, and when x is what value, △ oce ∽△ obc;

② Under the condition of ①, explore whether there is a point P on the parabola symmetry axis to make△ △PEM an isosceles triangle, and if so, find the coordinates of the point P; If it does not exist, please explain why.

Exercise: The copyright belongs to Jinyuan Math Studio and cannot be reproduced.

1.(20 12 Guangxi Baise 10) As shown in the figure, in the plane rectangular coordinate system, the parabola y = AX2+BX+6 passes through point A (-3,0) and point B (2,0). The straight line y=h(h (h is a constant, 0 < h < 6.

(2) When connecting BE and finding the value of H, the area of △BDE is the largest;

(3) A certain point m (-2,0) is known. Q: Is there such a straight line Y = H that △OMF is an isosceles triangle? If yes, find the value of H and the coordinates of G point; If it does not exist, please explain why.

y=h

2.(20 12 Jiangxi province 10 points) As shown in the figure, it is known that the quadratic function L 1: y = x2-4x+3 intersects with the X axis at two points A and B (point A is on the left of point B) and intersects with the Y axis at point C (1).

(2) Study the quadratic function L2: y = kx2-4kx+3k (k ≠ 0).

(1) Write two identical properties of images related to quadratic function L2 and quadratic function L 1;

② Is there a real number k that makes △ABP an equilateral triangle? If it exists, request the value of k; If it does not exist, please explain the reason; ③ If the straight line y=8k intersects with parabola L2 at points E and F, does the length of line segment EF change? If not, request the length of EF; If yes, please explain why.

3.(20 12, Hengyang, Hunan 10 minutes) As shown in the figure, it is known that the vertex of parabola is the coordinate origin O, the vertices A and D of rectangular ABCD are on parabola, AD is parallel to X axis and intersects Y axis at point F, the midpoint E of AB is on X axis, and the coordinate of point B is (2, 1).

(1) Find the analytical expression of this parabola. (2) The intersection point P is perpendicular to the straight line where CB is located, and the vertical foot is point R. ① Verification: PF = PR② Is there a point P that makes △PFR an equilateral triangle? If it exists, find the coordinates of point P; If it does not exist, please explain the reason; ③ Extend the intersecting parabola of PF to another point Q, where the crossing point Q is the vertical line of BC and the vertical foot S, and try to judge the shape of △RSF.

4.(20 12, Yongzhou, Hunan, 10) As shown in the figure, it is known that the image of quadratic function Y = AX2+BX- 1 (A ≠ 0) passes through points A (2 2,0) and B (4 4,3), and L is passed.

(1) Find the analytical formula of the quadratic function y = AX2+Bx- 1 (a ≠ 0); (2) Please directly write out the range of values corresponding to X to make y < 0; (3) When m=0, m=2 and m=4, calculate the values of |PO|2 and |PH|2 respectively, observe the law, guess a conclusion, and prove that this conclusion is true for any real number m;

(4) Is there a real number m that can make △POH a regular triangle? If it exists, find the value of m; If it does not exist, please explain why.

5.(20 12 Guangdong Meizhou 1 1) As shown in the figure, in the right-angle OABC A (6,0), C (0 0,2) and D (0 0,3), the light L passes through point D and is parallel to the X axis, and points P and Q are the L and X axes respectively.

(1) ① The coordinates of point B are; ②∠CAO= degree; ③ When point Q coincides with point A, the coordinate of point P is: (write the answer directly)

(2) Let the center of OA be n, PQ and line segment AC intersect at point M, is there a point P that makes △AMN an isosceles triangle? If it exists, please write directly that the abscissa of point P is m; If it does not exist, please explain why.

(3) Let the abscissa of point P be X, and the overlapping area of △OPQ and right-angle OABC be S. Try to find the functional relationship between S and X and the range of the corresponding independent variable X. 。

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Example 1: (20 12 Zaozhuang, Shandong 10) In the plane rectangular coordinate system, now put an isosceles right triangle ABC in the second quadrant and recline.

On the two coordinate axes, point C is (-1, 0). As shown in the figure, point B is on the parabola Y = x2+x-2 image, and it passes through point B..

BD⊥x axis, the vertical foot is D, and the abscissa of point B is -3.

(1) Verification: △ BDC △ COA;

(2) Find the functional relationship on which BC is online;

(3) Is there a point P on the parabola axis of symmetry, so that △ACP is a right triangle with AC on the right? If it exists, find out what it is.

Use a small p coordinate; If it does not exist, please explain why.

Example 2: (20 12 Chongqing 12) As shown in the figure, in the right-angled trapezoidal ABCD, AD∥BC, ∠ B = 90, AD=2, BC=6, AB = 3. E is a point on the side of BC. Make a square BEFG with BE as the side.

(1) Find the length of BE when the vertex f of the square just falls on the diagonal AC;

(2) Translate the square B'EFG( 1) in question to the right along BC. Note that the square BEFC in translation is a square B'EFG, and stop translating when point E coincides with point C. Let the translation distance be t, and the side EF of the square B'EFG intersects with AC at point M to connect B'D, B'M and DM. Is there such a t that if △ exists, find the value of t; If it does not exist, please explain the reason;

(3) In the translation process of question (2), let the area of the overlapping part of square B'EFG and △ADC be S, please directly write the functional relationship between S and T and the range of independent variable T. 。

Example 3: (20 12, Chifeng, Inner Mongolia, 12) As shown in the figure, the parabola intersects with the X axis at two points A and B (point A is on the left of point B), intersects with the Y axis at point C, points C and F are symmetrical about the parabola, and the straight line AF intersects with the Y axis at point E, | oc |: | OA | = 5: 65438

(1) Find the analytical formula of parabola;

(2) Find the analytical formula of straight line AF;

(3) Is there a point P on the straight line AF, so that △CFP is a right triangle? If it exists, find out the coordinates of point P; If it does not exist, explain why.

Example 4: (20 12 Hainan Province 13) As shown in the figure, a quadratic function image with vertex P(4, -4) passes through the origin (0,0) and point A is on the image.

OA intersects its symmetry axis at point m, and points m and n are symmetrical about point p, connecting an and on.

(1) Find the relation of this quadratic function.

(2) If the coordinate of point A is (6, -3), find the area of △ANO.

(3) When point A moves on the quadratic function image on the right side of the symmetry axis, please answer the following questions:

① Proof: ∠ANM=∠ONM

②△②△ANO can be a right triangle? If yes, request the coordinates of all qualified points A, if not, please explain the reasons.

Exercise: The copyright belongs to Jinyuan Math Studio and cannot be reproduced.

1.(20 12, Hechi, Guangxi 12) As shown in the figure, in the isosceles triangle ABC, AB=AC, with the middle vertical line and BC as the base.

Establish a plane rectangular coordinate system on a straight line, and the parabola passes through point A and point B. 。

(1) Write the coordinates of point A and point B;

(2) If the straight line L coincident with the Y axis moves to the right at a speed of 2 unit lengths per second, the line segments OA, CA and parabola intersect respectively.

The line connects PA and PB at points e, m and p. Let the moving time of the straight line L be t (0 < t < 4) seconds, find the functional relationship between the area s (area unit) and t (second) of the quadrilateral PBCA, and find the maximum area of the quadrilateral PBCA.

(3) Under the condition of (2), is there a point P on the parabola that makes △PAM a right triangle? If it exists, request point p.

Coordinates of; If it does not exist, please explain why.

2: (20 12, Shaoyang, Hunan, 12) As shown in the figure, the straight line intersects with the X axis at point A (4,0), intersects with the Y axis at point B, and folds △AOB along the Y axis to make point A fall on the X axis, and the corresponding point of point A is point C. 。

(1) Find the coordinates of point C;

⑵ Set point P as the moving point on the line segment CA, which does not coincide with points A and C, connect PB, and make the ray PM and AB intersect at point M with point P as the endpoint, then ∠BPM=∠BAC① can be proved as △ PBC ∠△ MPa;

② Is there a point P that makes △PBM a right triangle? If it exists, request the coordinates of point P; If it does not exist, please explain why.

3.(20 12, Yunnan Province, 9 points) As shown in the figure, in the plane rectangular coordinate system, the straight line intersects with the X axis at point P, intersects with the Y axis at point A, and the parabolic image passes through point E (-1, 0) and intersects with the straight line at points A and B. 。

(1) Find the analytical formula (relation) of parabola; (2) After passing through point A, make AC⊥AB pass through X axis at point C, and find the coordinates of point C;

(3) Are there m points on the coordinate axis besides point C, so that △MAB is a right triangle? If it exists, request the coordinates of point m; If it does not exist, please explain why.

Typical example: The copyright belongs to Jinyuan Mathematics Studio and cannot be reproduced.

Example 1: (20 12 Shanxi Province 14) Synthesis and exercise: As shown in the figure, in the plane rectangular coordinate system, the parabola y =-x2+2x+3 intersects with the X axis at points A and B, intersects with the Y axis at point C, and point D is the vertex of the parabola.

(1) Find the analytical formula of two-point coordinates of straight line AC and B.D.;

(2) Point P is a moving point on the X-axis, and the straight line l∑AC intersects the parabola at point Q. Try to explore: With the movement of point P, is there a point Q on the parabola, making the quadrilateral with points A, P, Q and C as its vertices a parallelogram? If it exists, please write the coordinates of the qualified point Q directly; If it does not exist, please explain why.

(3) Please find a point m on the straight line AC to minimize the perimeter of △BDM, and find the coordinates of the point m. 。

Example 2: (20 12 Shandong Rizhao 10) As shown in the figure, the image of quadratic function y = x2+bx+c intersects with the X axis at point A and point B, and the coordinates of point A are

(-3,0), a straight line passing through point B intersects a parabola at point D (-2,3).

(1) Find the analytical expressions of parabola and straight line BD;

(2) A straight line EF∨BD passing through point E(a, 0) on the X axis (point E is on the right side of point B) intersects a parabola at point F. Is there a real number A that makes the quadrilateral BDFE a parallelogram? If it exists, find the qualified; If it does not exist, please explain why.

Example 3: (20 12 Guangxi Beihai 12) As shown in the figure, there are Rt△AB=AC, ∠ A = 90, AB=AC, A (-2,0), B (0, 1) in the plane rectangular coordinate system.

(1) Find the value of d;

(2) Translate △ABC along the positive direction of the X axis, and the points B' and C' corresponding to points B and C in the first quadrant just fall on an inverse proportional function image. Find the analytical expression of this inverse proportional function and straight line B'C' at this time;

(3) Under the condition of (2), the straight line B'C' intersects with the Y axis at point G, and it is asked whether there are points m on the X axis and p on the inverse proportional function image to make the quadrilateral PGMC' a parallelogram. If yes, request the coordinates of point M and point P; If it does not exist, please explain why.

Example 4: (20 12, Dandong, Liaoning, 14) It is known that the parabola intersects with the Y axis at point C, and intersects with the X axis at points A and B. The coordinate of point A is (-1, 0), and O is the coordinate origin, and.

(1) Find the function expression of parabola; (2) Write the function expression of straight line BC directly;

(3) As shown in figure 1, d is a point on the negative semi-axis of Y axis, and OD=2. Make a square ODEF whose side length is the outer diameter. Move the square ODEF along the positive direction of the X axis at a speed of 1 unit per second. During the movement, let the area of the overlapping part of the square ODEF and △OBC be s, and the movement time be t seconds (0 < t ≤ 2.

It is found that: ① the functional relationship between S and T;

② Is there a maximum value of S during exercise? If it exists, write this maximum value directly; If it does not exist, please explain why.

(4) As shown in Figure 2, point P( 1, k) is on the straight line BC, point M is on the X axis, and point N is on the parabola. Is there a parallelogram with vertices a, m, n and p? If it exists, please write down the coordinates of point M directly; If it does not exist, please explain why.

Example 5: (20 12 Heilongjiang Heihe, Qiqihar, Daxinganling, Jixi 10) As shown in the figure, in the plane rectangular coordinate system, it is known that the two right-angled sides 0A and 08 of Rt△AOB are on the Y axis and the X axis respectively, and the lengths of OA and OB are the equation X2-7x+12 = 0 (. 0B), the moving point P starts from the point A and moves to the point O at the speed of L unit lengths per second on the line segment AO; At the same time, the moving point Q starts from point B and moves to point A at a speed of 2 unit lengths per second on the line BA, and the moving time of point P and point Q is t seconds.

(1) Find the coordinates of point A and point B (2) When t is what value, △APQ is similar to △AOB, and directly write the coordinates of point Q at this time.

(3) When t=2, is there a point m in the coordinate plane, which makes the quadrilateral with vertices A, P, Q and M a parallelogram? If it exists, please write the coordinates of point M directly; If it does not exist, please explain why.

Exercise: The copyright belongs to Jinyuan Math Studio and cannot be reproduced.

1.(20 12, Anshun, Guizhou, 14 minutes) As shown in the figure, in the plane rectangular coordinate system xOy, the side lengths OA and OC of the rectangular OABC are 12cm and 6cm, respectively, and the points A and C are on the negative semi-axis of the Y axis and the positive semi-axis of the X axis, respectively, and the parabola Y =

(1) Find the analytical formula of parabola.

(2) If point P starts from point A, it moves along AB side to end point B at a speed of 1cm/s, and point Q starts from point B and moves along BC side to end point C at a speed of 2 cm/s. 。

① At the t second after exercise, let the area of △PBQ be S, try to write the functional relationship between S and T, and write the value range of T. 。

② When S reaches the maximum, is there a point R on the parabola, so that the quadrilateral with P, B, Q and R as its vertices is a parallelogram? If it exists, find the coordinates of point R; If it does not exist, please explain why.

2.(20 12, Enshi, Hubei, 8 points) As shown in the figure, it is known that the parabola y=﹣x2+bx+c intersects a straight line at two points A (﹣ 1, 0) and C (2, 3), and intersects with the Y axis at point N, and its vertex is

(2) Set point M(3, m) to find the value of m when the value of MN+MD is minimum;

(3) If the symmetry axis of parabola intersects with the straight line AC at point B, E is any point on the straight line AC, the passing point E is EF∨BD and the parabola intersects with point F, can the quadrilateral with vertices B, D, E and F be a parallelogram? If yes, find the coordinates of point e; If not, please explain the reasons; (4) If p is the moving point on the parabola above the straight line AC, find the maximum area of △APC.

(2012 Yibin, Sichuan 10) As shown in the figure, the vertex A of the parabola y = x2-2x+c is on the straight line l: y = x-5.

(1) Find the coordinates of the vertex A of the parabola;

(2) Let the parabola intersect the Y axis at point B and the X axis at point C. D (point C is to the left of point D), and try to judge the shape of △ABD;

(3) Is there a point P on the straight line L, so that the quadrilateral with points P and A.B.D as vertices is a parallelogram? If it exists, find the coordinates of point P; If it does not exist, please explain why.

4.(20 12 Loudi, Hunan 10) It is known that the image of quadratic function y = x2-(m2-2) x-2m intersects with the X axis at points A (x 1 0) and B (x2, 0), x660.

(1) Find the analytic expression of this quadratic function;

(2) Inquiry: Is there a point P on the straight line y=x+3, which makes the quadrilateral PACB a parallelogram? If yes, find the coordinates of point P; If not, please explain why.

Fourth, there are problems with rectangles, diamonds and squares;

Typical example: The copyright belongs to Jinyuan Mathematics Studio and cannot be reproduced.

Example 1: (BC= 12 Longdong, Heilongjiang 10) As shown in the figure, in the plane rectangular coordinate system, the edges OC and OA of the right-angled trapezoidal OABC coincide with the X axis and the Y axis respectively, and AB∥OC, ∠ AOC = 90, ∠ BCO.

(2) If the straight line DE intersects with the trapezoidal diagonal BO at point D, and intersects with the Y axis at point E, and OE=4, OD=2BD, find the analytical formula of the straight line DE; (3) If the point P is the moving point on the straight line DE in (2), is there a point Q on the coordinate plane, so that the quadrilateral with vertices O, E, P and Q is a diamond? If it exists, please write the coordinates of Q point directly; If it does not exist, please explain why.

Example 2: (20 12, Liupanshui, Guizhou, 16) as shown in figure 1, it is known that in △ABC, AB= 10cm, AC=8cm and BC = 6 cm. If point P starts from B and moves at a uniform speed in the direction of BA, point Q starts from A and moves at a uniform speed in the direction of AC to point C,

(1) When t is what value, PQ∑BC.

(2) Let the area of △AQP be S (unit: cm2), and when t is what value, S takes the maximum value to find the maximum value.

(3) Is there a moment t at which the straight line PQ exactly bisects the area of △ABC? If it exists, find the value of t at this time; If it does not exist, please explain why. (4) As shown in Figure 2, fold △AQP along AP to obtain quadrilateral AQPQ'. So, is there a time t that makes the quadrilateral AQPQ' a diamond? If it exists, find out the area of the diamond at this time; If it does not exist, please explain why.

Example 3: (20 12 Tieling, Liaoning 14) As shown in the figure, it is known that the parabola passes through the origin O and a point A (4 4,0) on the X axis, and the vertex of the parabola is E,

Its symmetry axis intersects with the X axis at point D, and the straight line passes through a point B (-2, m) on the parabola, intersects with the Y axis at point C, and intersects with the parabola.

The axis of symmetry intersects with point F.

(1) Find the analytical formula corresponding to the value of m and parabola;

(2)P is a point on a parabola, and if S△ADP=S△ADC, the coordinates of all qualified points p are found;

(3) The point Q is any point on the plane, and the point M starts from the point F and moves along the axis of symmetry at a constant speed of 1 unit length per second. If the movement time of point M is t seconds, can the quadrilateral with four vertices of Q, A, E and M be made into a diamond? If yes, please directly write the value of the movement time t of point M; If not, please explain why.

Standby chart

Example 4: (20 12, Zhangzhou, Fujian 12) Known parabola y= x2+ 1 (as shown in the figure).

(1) Fill in the blank: the vertex coordinates of the parabola are (_ _ _ _ _ _, _ _ _ _ _) and the symmetry axis is _ _ _ _ _ _ _ _;

(2) The point A (0 0,2) on the Y axis is known, the point P is on a parabola, the passing point P is PB⊥x axis, and the vertical foot is B. If △PAB is an equilateral triangle, find the coordinates of the point P;

(3) Under the condition of (2), the point m is on the straight line AP. Is there a point n on the plane that makes the quadrilateral OAMN a diamond? If it exists, directly write the coordinates of all points n that meet the conditions; If it does not exist, please explain why.

Example 5: (20 12, Tongliao 12, Inner Mongolia) As shown in the figure, in the plane rectangular coordinate system, a square ABCD is placed in the first quadrant and leans on two coordinate axes, with point A (0 0,2), point B (1 0) and parabola Y = AX2.

(1) Find the coordinates of point C; (2) Find the analytical formula of parabola; The copyright belongs to Jinyuan Mathematics Studio and cannot be reproduced.

(3) Are there points P and Q on the parabola (except points C and D) to make the quadrilateral ABPQ square? If point P and point Q have two coordinates, if not, explain the reason.

Exercise: The copyright belongs to Jinyuan Math Studio and cannot be reproduced.

1.(20 12 minutes) As shown in the figure, in the plane rectangular coordinate system, three vertices of rectangular ABCD are known: B (1 0), C (3 3,0) and D (3 3,4). A parabola with vertices is Y = AX2+.

(1) Write the coordinates of point A directly and find the analytical formula of parabola;

(2) If the intersection e is EF⊥AD at F and the parabola at G, what is the value of t and the area of △ACG is the largest? What is the maximum value? (3) In the process of moving points P and Q, when the value of t is what, there is a point H in the rectangle ABCD (including the boundary), so that the quadrilateral with the vertices of C, Q, E and H is a diamond? Please write the value of t directly.

2.(20 12, Fuzhou, Fujian, 13) as shown in figure 1, in Rt△ABC, ∠ c = 90? , AC = 6, BC = 8, moving point P moves from point A to point C at a speed of/kloc-0 per unit length per second, moving point Q moves from point C to point B along CB at a speed of 2 unit lengths per second, intersection point P is PD∑BC, and intersection point AB connects PQ. Point p and point q start from point a and point c respectively at the same time, when

(1) is directly expressed by an algebraic expression t: QB = _ _ _ _ _ _, PD = _ _ _ _ _ _

(2) Is there a t value that makes the quadrilateral PDBQ a diamond? If it exists, find the value of t; If it doesn't exist, explain the reason, and discuss how to change the speed of Q point (uniform motion) to make the quadrilateral PDBQ diamond at a certain moment, and find the speed of Q point; (3) As shown in Figure ②, in the whole movement process, find the path length of the midpoint m of the line segment PQ.

3.(20 12, Jinzhou, Liaoning, 14 minutes) As shown in the figure, the parabola and the axis intersect at point C, the straight line L is the parabola symmetry axis, and the point P is in the third quadrant, which is the vertex of the parabola. The distance from p to the axis is 1. The symmetrical point of point C about line L is A, which connects AC intersecting lines L and B. 。

(1) Find the expression of parabola; The copyright belongs to Jinyuan Mathematics Studio and cannot be reproduced.

(2) The straight line intersects the parabola at point D, intersects the axis at point F, and connects BD at point E in the first quadrant, and

DE:BE=4: 1。 Find the expression of straight line;

(3) If n is a point in the plane rectangular coordinate system, is there a point M on the straight line so that points O, F, M and N are

The quadrilateral of the vertex is a diamond? If it exists, directly write the coordinates of point m; If it does not exist, please explain why.

4.(20 12 Qinghai Province 12) As shown in the figure, in the plane rectangular coordinate system, the image of the quadratic function y=x2+bx+c intersects with the X axis at two points A and B, with point A on the left side of the origin, and the coordinate of point B at (3,0), and intersecting with the Y axis at point C (0,

(2) Connect PO and PC, and fold △POC along Co to get quadrilateral POP'C. Is there a point P that makes quadrilateral POP' C a diamond? If yes, request the coordinates of point P at this time; If it does not exist, please explain why.

(3) When the point P moves to what position, the area of the quadrilateral ABPC is the largest, and the coordinates of the point P and the maximum area of the quadrilateral ABPC are obtained.