Question

На стороне BC выпуклого четырехугольника ABCD отмечена точка P, а вне четырехугольника — точка Q. При этом отрезки BP и AQ, а также отрезки CP и DQ имеют общие середины. Докажите, что отрезки AC и BD имеют общую середину.

Answer 1

Ответ:

Объяснение:На стороне BC выпуклого четырехугольника ABCD

Answer 2

Proof that segments AC and BD have a common midpoint

To prove that segments AC and BD have a common midpoint, we can use the given information that segments BP and AQ, as well as segments CP and DQ, have common midpoints.

Let's denote the common midpoint of segments BP and AQ as M1, and the common midpoint of segments CP and DQ as M2.

To prove that segments AC and BD have a common midpoint, we need to show that the midpoint of segment AC is the same as the midpoint of segment BD.

Let's denote the midpoint of segment AC as M3 and the midpoint of segment BD as M4.

To prove that M3 and M4 are the same point, we can use the concept of collinearity. If M3 and M4 are the same point, then they must lie on the same line.

We can consider the line passing through points A and C. Since M1 is the midpoint of segment BP, it lies on the line passing through points B and P. Similarly, since M2 is the midpoint of segment CP, it lies on the line passing through points C and P.

Now, let's consider the line passing through points B and D. Since M1 is the midpoint of segment AQ, it lies on the line passing through points A and Q. Similarly, since M2 is the midpoint of segment DQ, it lies on the line passing through points D and Q.

Since M1 lies on both the line passing through points B and P and the line passing through points A and Q, it must be the intersection point of these two lines. Similarly, since M2 lies on both the line passing through points C and P and the line passing through points D and Q, it must be the intersection point of these two lines.

Therefore, M1 and M2 are the same point, which we can denote as M.

Now, let's consider the line passing through points A and C. Since M is the intersection point of the lines passing through points B and P and the lines passing through points A and Q, it must also lie on the line passing through points A and C.

Similarly, let's consider the line passing through points B and D. Since M is the intersection point of the lines passing through points C and P and the lines passing through points D and Q, it must also lie on the line passing through points B and D.

Therefore, M is the common midpoint of segments AC and BD.

In conclusion, we have proved that segments AC and BD have a common midpoint, which is denoted as M.

Note: The proof provided above is based on the given information and the concept of collinearity.