Category Archives: Geometry

A MATHCOUNTS Problem with Multiple Answers 🙂

Posted on August 19, 2024 by kevin

The following is MATHCOUNTS 2012 National Competition Sprint Round Problem 22: In circle $O$, shown, $OP=2$ units, $PL=8$ units, $PK=9$ units and $NK=18$ units. Points $K$, $P$ and $M$ are collinear, as are points $L$, $P$, $O$ and $N$. What … Continue reading →

Posted in Fun Math Facts, Geometry, MATHCOUNTS, Trigonometry | Comments Off on A MATHCOUNTS Problem with Multiple Answers 🙂

Geometry Probability – 1

Posted on July 22, 2024 by kevin

3 random points are chosen independently on a circle. What is probability that they form an acute triangle? Click here for the solution. Solution: This problem is equivalent to the following: Two cuts are randomly made on a line of … Continue reading →

Posted in Algebra, Geometry | Comments Off on Geometry Probability – 1

Algebra/Geometry Challenge – 1

Posted on June 17, 2024 by kevin

As shown in the following figure, $D$, $E$, $F$ are on the sides of $\triangle{ABC}$, $AC$, $AB$, and $BC$ respectively. $AE=BE$, $AD=6$, $CD=7$, $BF=2$, $CF=9$. $DEFG$ is a square. The length of $AB$ can be expressed as $\ \ \ … Continue reading →

Posted in Algebra, Geometry | Comments Off on Algebra/Geometry Challenge – 1

Circles in a Square – Part 12

Posted on May 21, 2024 by kevin

As shown in the figure, $ABCD$ is a square, $E$ is the mid-point of $AB$. The circle with its center at $H$ is tangent with $AD$, $AE$ and $DE$. The circle with its center at $F$ is tangent with $BC$, … Continue reading →

Posted in Circles in a Square, Geometry | Comments Off on Circles in a Square – Part 12

Geometry Challenge – 16

Posted on April 4, 2024 by kevin

$O$ is an interior point of regular hexgon $ABCDEF$. Prove that $[\triangle{OEF}]=2[\triangle{OAB}]+2[\triangle{OCD}]-3[\triangle{OBC}]$ Click here for the proof. Proof 1: Without loss of generality, assume that $ABCDEF$ is unit regular hexgon with its center at $(0,0)$, $A$ at $(-\dfrac{1}{2},-\dfrac{\sqrt{3}}{2})$, $B$ at … Continue reading →

Posted in Geometry | Comments Off on Geometry Challenge – 16