Segment CD intersects plane β, point E is the midpoint of CD. Parallel straight lines are drawn through points

Segment CD intersects plane β, point E is the midpoint of CD. Parallel straight lines are drawn through points C, D and E, intersecting the plane β, respectively, at points C1, D1, and E1. Find EE1 if CC1 = 6 / √3 cm and DD1 = √3 cm.

Decision.
Let the segment CD intersect the plane β, the point E is the midpoint of CD, and parallel lines are drawn through the points C, D, and E, intersecting the plane β, respectively, at the points C₁, D₁, and E₁. Through the point D К in the plane ABDём draw a straight line КD₁ | | CD. Let M be the intersection points of the lines KD₁ and EE₁. Then, in the resulting triangle C₁D₁K, the sides CK = CC₁ + DD₁, the middle line EM = EE₁ + DD₁ (by Thales’ theorem, KM = M D₁). By the property of the middle line of the triangle, EM = CK / 2. It is known from the problem statement that CC₁ = 6 / √3 cm and DD₁ = √3 cm, then we find ЕE₁ = ЕМ – DD₁; EE₁ = CK / 2 – DD₁; EE₁ = (CC₁ + DD₁) / 2 – DD₁. Substitute the values ​​of the quantities and make calculations: EE₁ = (6 / √3 + √3) / 2 – √3 = 0.5 ∙ √3 (cm).
Answer: 0.5 ∙ √3 cm.