Riemann surface
matlab

Plane 1.1. The bases of a trapezoid are $a$ and $b$. Find the length of the segment that the diagonals of the trapezoid intersept on the trapezoid’s midline.

Solutions
1.1. a) Let $P$ and $Q$ be the midpoints of $A B$ and $C D$; let $K$ and $L$ be the intersection points of $P Q$ with the diagonals $A C$ and $B D$, respectively. Then $P L=\frac{a}{2}$ and $P K=\frac{1}{2} b$ and so $K L=P L-P K=\frac{1}{2}(a-b)$.
b) Take point $F$ on $A D$ such that $B F | C D$. Let $E$ be the intersection point of $M N$ with $B F$. Then
$$M N=M E+E N=$$
$$\frac{q \cdot A F}{p+q}+b=\frac{q(a-b)+(p+q) b}{p+q}=\frac{q a+p b}{p+q}$$

1.2. Prove that the midpoints of the sides of an arbitrary quadrilateral are vertices of a parallelogram. For what quadrilaterals this parallelogram is a rectangle, a rhombus, a square?

1.2. Consider quadrilateral $A B C D$. Let $K, L, M$ and $N$ be the midpoints of sides $A B$, $B C, C D$ and $D A$, respectively. Then $K L=M N=\frac{1}{2} A C$ and $K L | M N$, that is $K L M N$ is a parallelogram. It becomes clear now that $K L M N$ is a rectangle if the diagonals $A C$ and $B D$ are perpendicular, a rhombus if $A C=B D$, and a square if $A C$ and $B D$ are of equal length and perpendicular to each other.

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