Category Archives: Geometry

Geometry Probability – 4

Posted on February 2, 2025 by kevin

Let $D$ is an interior point inside equilateral $\triangle{ABC}$. Find the probability that the line segments of $AD$, $BD$, and $CD$ are the side of: (1) a triangle, (2) a right triangle, (3) an obtuse triangle, and (4) an acute … Continue reading →

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Geometry Challenge – 17

Posted on January 12, 2025 by kevin

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 →

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Algebra/Geometry Challenge – 2

Posted on December 8, 2024 by kevin

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 →

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Stengel’s Theorem – Extended Cross Ladders Theorem

Posted on November 7, 2024 by kevin

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 →

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Geometry Challenge – Area of Irregular Pentagon

Posted on October 15, 2024 by kevin

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 →