Geometry
Problems related to shapes, spaces, measurements, etc.
Problems in this category
The diagonals of a rhombus $ABCD$ are $\vec{AC}$ and $\vec{BD}$, and they intersect at point $O$. Ex...
VectorsGeometryRhombusVector AdditionVector Subtraction
2025/3/27
Given the vectors $\vec{OM} = (3, 1)$ and $\vec{ON} = (-5, -1)$, find the vector $\frac{1}{2}\vec{MN...
VectorsVector OperationsCoordinate Geometry
2025/3/27
Let $A_1, B_1, C_1$ be the midpoints of the sides $BC, CA, AB$ of triangle $ABC$ respectively. Let $...
VectorsTriangle GeometryMidpoint
2025/3/27
Let $A_1, B_1, C_1$ be the midpoints of sides $BC, CA, AB$ of triangle $ABC$, respectively. Let $M$ ...
VectorsTriangle GeometryMidpointsVector AdditionGeometric Proof
2025/3/27
The problem asks us to simplify the expression $\vec{AB} - \vec{AC} + \vec{BD} - \vec{CD}$.
VectorsVector AlgebraVector AdditionVector Subtraction
2025/3/27
The problem states that two angles of a triangle are $60^\circ$ and $72^\circ$. We need to find the ...
TrianglesAnglesAltitudesOrthocenter
2025/3/27
Let $T$ be the centroid of triangle $ABC$. Prove that the vector sum $\vec{TA} + \vec{TB} + \vec{TC}...
VectorsTrianglesCentroidGeometric Proof
2025/3/27
The problem asks us to prove something about triangle $ABC$ given the equation $\vec{TA} + \vec{TB} ...
VectorsTrianglesCentroidGeometric Proofs
2025/3/27
The angle bisectors of triangle $ABC$ intersect the circumcircle of triangle $ABC$ at points $X, Y, ...
Triangle GeometryAngle BisectorsCircumcircleInscribed Angles
2025/3/27
The problem asks to calculate the area of a regular polygon circumscribed about a circle, given that...
PolygonsAreaPerimeterCircumscribedLimits
2025/3/27