As shown in the figure, the vertex ab of equilateral triangle ABC with side length of 6 and ∠ mon = 60 ° is on OM and on respectively, and point P is the intersection of bisector of ∠ BAC and ∠ ABC

It is proved that the triangle Pax and triangle PBY are congruent, so PX = py, so p is on the angle bisector. (congruent proof: PA = Pb, X and y are perpendicular feet, so we only need to prove ∠ Pax = ∠ PBY, and ∠ Pax = ∠ Pao = ∠ Bao + 30 °, PBY = ∠ CBY + 30 ° according to the external angle theorem, ∠

RELATED INFORMATIONS

1. As shown in the figure, given that point a is a point in the acute angle ∠ Mon, try to determine point B and point C on OM and on respectively, so as to minimize the perimeter of △ ABC. Write down the main steps of your drawing and mark the points you determine______ (required to draw a sketch and keep traces)
2. As shown in the figure, it is known that point a is a point in the acute angle ∠ mon. Try to determine point B and point C on OM and on respectively to minimize the perimeter of △ ABC As shown in the figure, it is known that point a is a point in the acute angle ∠ mon. Try to determine point B and point C on OM and on respectively to minimize the perimeter of △ ABC, and explain the reason
3. 23. As shown in the figure, it is known that point a is a point in the acute angle ∠ mon. Try to determine point B and point C on OM and on respectively, so as to minimize the perimeter of △ ABC
4. As shown in the figure, it is known that point a is a point in the acute angle ∠ mon. Try to determine point B and point C on OM and on respectively, so as to minimize the perimeter of △ ABC. Explain the reason
5. As shown in the figure, it is known that a is a point in the acute angle mon. Try to determine points B and C on OM and on respectively to minimize the perimeter of △ ABC, and explain the reason
6. Let a vertex of △ ABC be a (3, - 1), the bisector equations of ∠ B and ∠ C be x = 0 and y = x respectively, and then the equation of the straight line BC is obtained
7. Triangle ABC, a (5, - 2), B (7,3) are known, and the midpoint m of AC is on the Y axis, and the midpoint n of BC is on the X axis. Q: 1. The coordinates of vertex C; 2. The equation of line Mn
8. Given the two vertices a (4.7) B (- 2.6) of the triangle ABC, find the coordinates of point C such that the midpoint of AC is on the x-axis and the midpoint of BC is on the x-axis
9. Given two vertices a (- 3,7), B (2,5) of triangle ABC, the midpoint of tangent AC is on the x-axis, and the midpoint of BC is on the y-axis, then the coordinate of vertex C is?
10. If the midpoint of AC is on the x-axis and the midpoint of BC is on the y-axis, then the coordinate of C is () A. （2，-7）B. （-7，2）C. （-3，-5）D. （-5，-3）
11. Given ∠ AOB and ray OC, OM and on divide ∠ AOC and ∠ BOC equally. (1) if OC is outside ∠ AOB, try to explore the relationship between ∠ mon and ∠ AOB. (Fig. 2) (2) if OC is inside ∠ AOB, what is the relationship between ∠ mom and ∠ AOB? (Figure 1)
12. If OC is outside the angle AOB, try to explore the relationship between the angle mon and the angle AOB. (2) if OC (2) if OC is in angle AOB, what is the relationship between angle mon and angle AOB?
13. As shown in the figure, the coordinate of the quadratic function image passing through ABC three points a is (- 1,0), the coordinate of point B is (4,0), point C is on the positive half axis of Y axis and ab = OC
14. As shown in Figure 1, in the plane rectangular coordinate system, it is known that △ ABC is an equilateral triangle, the coordinate of point B is (12,0), and the moving point P moves from point a to point B on the line AB at a speed of units per second, and the movement time is set as T seconds. Take point P as the vertex, make equilateral △ PMN, and the points m and N are on the X axis (1) When t is the value, point m coincides with point o; (2) Find the P coordinate of the point and the side length of equilateral △ PMN (expressed by the algebraic expression of T); (3) If we take the midpoint D of ob, take od as the edge, make a rectangular odef as shown in Figure 2 in △ AOB, and point E is on line ab. let the area of the overlapping part of equilateral △ PMN and rectangular odef be s, request the functional relationship between S and T when 0 ≤ t ≤ 2 seconds, and find the maximum value of S Where's the test question? I want to know where this is from
15. In the plane rectangular coordinate system, the coordinates of ABC three points are (0,1), (2,0), (2,1.5). 1 1 point P (a, 1 / 2), use the formula containing a to express the area of the quadrilateral abop; 2. Under the condition of 1, is there a point P, so that the area of the quadrilateral ABOB is equal to the area of the triangle ABC? If there is, request the coordinates of point p; if not, please explain the reason
16. It is known that a parabola is symmetric about x-axis, its vertex is the origin of coordinates, and points P (2,4), a (x1, Y1), B (X2, Y2) are three points of the parabola. (I) find the parabola (1) Find the equation of the parabola (2) If the inclination angles of PA and Pb are complementary, the trajectory equation of the midpoint of AB is obtained (3) If ab ⊥ PA, find the planting range of the ordinate of point B
17. The parabola is symmetric about X axis, its vertex is at the origin of coordinates, and the points P (1,2), a (x1, Y1), B (X2, Y2) are all on the parabola The solution is "the parabola is symmetric about X axis, its vertex is at the origin of coordinates, and the points P (1,2), a (x1, Y1), B (X2, Y2) are all on the parabola, so the parabolic equation and its quasilinear equation can be solved."
18. As shown in the figure, the parabola is symmetric about X axis, its vertex is at the origin of coordinates, and the points P (1,2), a (x1, Y1), B (X2, Y2) are all on the parabola. (I) write out the equation of the parabola and its quasilinear equation; (II) when the slopes of PA and Pb exist and complement each other, find the value of Y1 + Y2 and the slope of line ab
19. The parabola with (1,2) as vertex intersects with X-axis at a and B, intersects with Y-axis at m, and the coordinate of a is (- 1,0). The area of △ AMB is calculated
20. It is known that the common point of the parabola y = ax ^ 2 = BX + C and X axis is a (- 1,0) B (3,0) and Y axis is C, and the vertex is d If the triangle ABC is a right triangle, try to find the value of A Q2: is there a non-zero constant a that makes ABCD on a circle? y=aX^2+bX+c