Question

Докажите, что отрезок, проходящий через середины оснований трапеции, проходит через пересечения диагоналей трапеции. Буду рад, если рисунок будет, а также, пожалуйста, привести доказательство без того факта, что диагонали перпендикулярны!

Answer 1

O - точка пересечения диагоналей.

Диагонали образуют на основаниях подобные треугольники, △AOD~△COB. Медианы OM и ON являются соответствующими элементами подобных треугольников и составляют равные углы с соответствующими сторонами, ∠AOM=∠CON. Лучи OA и OC составляют прямую, следовательно OM и ON также составляют прямую.

Answer 2

The Claim:

The claim is that the segment passing through the midpoints of the bases of a trapezoid also passes through the intersection of its diagonals.

Proof:

To prove this claim, let's consider a trapezoid ABCD, where AB and CD are the parallel bases, and E is the intersection point of the diagonals AC and BD. Let M and N be the midpoints of the bases AB and CD, respectively.

We need to show that the segment MN passes through the point E.

Step 1: Draw the trapezoid ABCD and its diagonals AC and BD.


Step 2: Draw the segment MN passing through the midpoints M and N of the bases AB and CD, respectively.


Step 3: Extend the segment MN to intersect the diagonals AC and BD at points F and G, respectively.


Step 4: To prove that the segment MN passes through the intersection point E of the diagonals AC and BD, we need to show that F and G coincide with E.

Step 5: Since M and N are the midpoints of the bases AB and CD, respectively, we can conclude that MN is parallel to the bases AB and CD.

Step 6: By the properties of a trapezoid, we know that the diagonals AC and BD intersect at a point E that divides them proportionally. Therefore, we can conclude that the triangles AEF and CEG are similar.

Step 7: Since the triangles AEF and CEG are similar, their corresponding angles are equal. Therefore, angle AFE is equal to angle CEG, and angle AEF is equal to angle CGE.

Step 8: Since MN is parallel to the bases AB and CD, we can conclude that angle AFE is equal to angle AEM and angle CEG is equal to angle CEN.

Step 9: From steps 7 and 8, we can conclude that angle AEM is equal to angle CEN.

Step 10: Since angle AEM is equal to angle CEN, and angle AME is equal to angle CNE (as they are vertical angles), we can conclude that the triangles AEM and CEN are similar.

Step 11: Since the triangles AEM and CEN are similar, their corresponding sides are proportional. Therefore, we can conclude that AE/CE = EM/EN.

Step 12: Since M and N are the midpoints of the bases AB and CD, respectively, we can conclude that EM = EN.

Step 13: From steps 11 and 12, we can conclude that AE = CE.

Step 14: Since AE = CE, we can conclude that F coincides with E.

Step 15: Similarly, we can prove that G coincides with E.

Conclusion: From steps 14 and 15, we have shown that the segment MN passing through the midpoints of the bases of a trapezoid also passes through the intersection of its diagonals.

Therefore, the claim has been proven.