Geometry
Problems related to shapes, spaces, measurements, etc.
Problems in this category
In triangle $ABC$, $BC > AC$. Point $D$ is on side $BC$ such that $AC = CD$. The angles $\alpha$ and...
Triangle GeometryAngle BisectorIsosceles TriangleAngle Chasing
2025/3/29
The angle bisectors of angles $\beta$ and $\gamma$ in triangle $ABC$ form an angle $\varphi$. The ra...
TriangleAngle BisectorsAnglesSystem of Equations
2025/3/29
In triangle $ABC$, the angle bisector $CD$ of angle $\gamma$ intersects side $AB$ at an angle $\varp...
TriangleAngle BisectorIsosceles TriangleAngle CalculationGeometric Proof
2025/3/29
The problem asks to find the angle at which the angle bisectors of the acute angles in a right trian...
TrianglesAngle BisectorsRight TrianglesAngles
2025/3/29
Given a regular hexagon $ABCDEF$, let $M$ be the midpoint of segment $DE$, $N$ be the midpoint of se...
VectorsGeometryHexagon
2025/3/29
Given vectors $\vec{a}$, $\vec{b}$, and $\vec{c}$ such that $\vec{a} + \vec{b} + \vec{c} = 0$, $|\ve...
VectorsDot ProductAngle between vectorsVector Algebra
2025/3/29
Given a square $ABCD$, where $CE$ is parallel to $BD$, and $E$ belongs to $AD$. a) Prove that $\vec{...
VectorsGeometrySquaresVector AdditionVector Proofs
2025/3/29
Given a regular hexagon $ABCDEF$, let $A_1, B_1, C_1, D_1, E_1, F_1$ be the midpoints of the sides $...
VectorsGeometryHexagonMidpointVector Addition
2025/3/29
Given a parallelogram constructed on vectors $\vec{AB}$ and $\vec{AD}$, express the vectors $\vec{OA...
VectorsParallelogramVector AdditionMidpoint
2025/3/29
Given a regular hexagon $ABCDEF$, with $\vec{AB} = \vec{p}$ and $\vec{BC} = \vec{q}$, express the ve...
VectorsGeometryHexagon
2025/3/29