Category Archives: Geometry

Math Olympiad Exercise – 2

Posted on May 23, 2021 by kevin

$\triangle{ABC}$ is a right triangle, with $\angle{ACB}=90^\circ$ and $\angle{BAC}=50^\circ$. Point $D$ and $E$ are on line $BC$ and $AC$ respectively, so that $\angle{BAD}=\angle{ABE}=30^\circ$. Connect $D$ and $E$, find the value of $\angle{ADE}$. Click here for the solution. Solution: Let $AD$ … Continue reading →

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LMT 2020 Fall – Problem 9

Posted on May 17, 2021 by kevin

$\triangle{ABC}$ has a right angle at $B$, $AB = 12$, and $BC = 16$. Let $M$ be the midpoint of $AC$. Let $ω_1$ be the incircle of $\triangle{ABM}$ and $ω_2$ be the incircle of $\triangle{BCM}$. The line externally tangent to … Continue reading →

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MATHCOUNTS Exercise – 22

Posted on January 1, 2021 by kevin

Point $E$ and $F$ are inside the square $ABCD$, with $DE=10$, $EF=6$, $BF=4$, and $\angle{DEF}=\angle{BFE}=90^\circ$. Find the area of the square $ABCD$. Click here for the solutions. Solution 1: Draw line $DB$, intersecting $EF$ at $G$. We have $\triangle{DEG}\sim\triangle{BFG}$. $\therefore … Continue reading →

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Geometry Fold Problems

Posted on October 16, 2020 by kevin

A unit equilateral $\triangle{ABC}$ is folded over line $DE$, forming a quadrilateral $BCDE$, with $A$ touching $BC$ at $A’$, and $\triangle{BA’E}$ is a right triangle. The area of $BCDE$ is __________. Answer: $\dfrac{17\sqrt{3}-27}{8}\approx 0.305608$ A quarter-circle $ABC$, with its center … Continue reading →

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MATHCOUNTS Exercises – 21

Posted on February 15, 2020 by kevin

The perimeter of rhombus $ABCD$ is 8 and its area is 2. Find the value of $\angle{ABC}$ in terms of $\pi$. Click here for the solution. Solution Draw line $AE$ perpendicular to $BC$ with $E$ on $BC$. Connect $A$ and $C$ to form two congruent $\triangle{ABC}$ and … Continue reading →