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Category Archives: Geometry
Geometry Challenge – 17
Let $D$ be an interior point inside equilateral $\triangle{ABC}$, so that $\angle{BDC}=150^\circ$. Prove that the line segment $AD$, $BD$ and $CD$ are the sides of a right triangle. Click here for the proof. Proof: Rotating $\triangle{ADC}$ counter-clock-wise around $C$ by … Continue reading
Posted in Geometry
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Algebra/Geometry Challenge – 2
Cyclic quadrilateral $ABCD$ has lengths $BC=CD=3$, $AB=5$, $AD=8$. What is the length of the shorter diagonal of $ABCD$? Click here for the solution. Solution: Let $AC$ and $BD$ intersect at $E$, $x=BD$, $y=AE$, $z=CE$, $AC=y+z$. Because $BC=CD$, $AC$ is angle … Continue reading
Stengel’s Theorem – Extended Cross Ladders Theorem
If you remember, the Classic Cross Ladders Theorem was used in one of the solutions to this MathCounts problem. The Extended Cross Ladders Theorem, also known as Stengel’s Theorem, applies to cross ladders in a triangle. Specifically, in $\triangle{ABC}$ as … Continue reading
Posted in Daily Problems, Geometry
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Geometry Challenge – Area of Irregular Pentagon
As shown in the diagram below, a star sign consists of five straight lines. It produces five triangles and a pentagon. If areas of five triangles are 3, 10, 7, 15, and 8 square unit respectively. Find the area of … Continue reading
Geometry Probability – 3
Two points are randomly and uniformly selected from the interior of a circle. The center of the circle and the two points joined together form a triangle. What is the probability that the triangle is acute? Click here for the … Continue reading
Posted in Algebra, Calculus, Geometry, Probability
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