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Problems Geometry Plane geometry Vectors The sum and the difference of vectors, the multiplication of a vector by a scalar. Properties

14–14 2.0

Prove that the point $X$ lies on the line $A B$ if and only if $\vec{O X} = t \vec{O A} + (1 - t) \vec{O B}$ for some $t$ and any point $O$ .

Problem #PRU-57710

Problems Geometry Plane geometry Vectors The sum and the difference of vectors, the multiplication of a vector by a scalar. Properties

14–14 2.0

Several points are given and for some pairs $(A, B)$ of these points the vectors $\vec{A B}$ are taken, and at each point the same number of vectors begin and end. Prove that the sum of all the chosen vectors is $\vec{0}$ .

Problem #PRU-108604

Problems Geometry Plane geometry Triangles Relationship of side lengths and angles of a triangle. Solving triangles. Ceva's theorem and Menelaus's theorem Calculus Integral Existence of a definite integral Vectors Law of polygon of vectors Similar triangles

13–14 3.0

On the sides $A B$ , $B C$ and $A C$ of the triangle $A B C$ points $P$ , $M$ and $K$ are chosen so that the segments $A M$ , $B K$ and $C P$ intersect at one point and $\vec{A M} + \vec{B K} + \vec{C P} = 0$ Prove that $P$ , $M$ and $K$ are the midpoints of the sides of the triangle $A B C$ .