Question

exerc: abcd is a pairn s is hex of (ab) & l is the symetric of d with respect to s. 1) draw a figure. 2) prove that hlbd is apan 3) prove that lb & c collinear & deduce that b zc

Explanation:

Step1: Analyze given information

We have a quadrilateral \(ABCD\), \(S\) is the mid - point of \(AB\) and \(L\) is the symmetric of \(D\) with respect to \(S\).

Step2: Prove \(HLBD\) is a parallelogram (assuming \(H\) is a relevant point, but since not defined clearly, we'll focus on the general approach)

Since \(S\) is the mid - point of \(AB\) and \(L\) is the symmetric of \(D\) with respect to \(S\), we know that by the property of symmetry, \(DS = SL\) and \(AS=SB\). In a quadrilateral, if the diagonals bisect each other, it is a parallelogram. Here, the line segment joining the mid - points of the diagonals (related to the construction) implies the parallelogram property.

Step3: Prove \(L\), \(B\), \(C\) are collinear

We can use vector methods or coordinate - free geometry concepts. Let \(\overrightarrow{AS}=\overrightarrow{SB}\) and \(\overrightarrow{DS}=\overrightarrow{SL}\). We know that \(\overrightarrow{AB}=\overrightarrow{DB}-\overrightarrow{DA}\). Since \(S\) is the mid - point of \(AB\), we can express vectors in terms of known points. By using the fact that in a parallelogram (from step 2) and vector addition/subtraction, we can show that \(\overrightarrow{LB}\) and \(\overrightarrow{BC}\) are related in such a way that \(L\), \(B\), \(C\) are collinear.

Answer:

The detailed geometric proofs for the parallelogram and collinearity can be completed using vector or coordinate - free geometry concepts as described above. For the parallelogram property, use the mid - point and symmetry properties to show diagonal bisection. For collinearity, use vector relationships between the points.