Author Archives: kevin

Geometry Challenge – 3 ⭐⭐⭐

Posted on July 23, 2021 by kevin

A fixed point $Q$ lies on the bisector of $\angle{P}$. A moving point $A$ lies on one side of $\angle{P}$, with line $AQ$ intersecting the other side of $\angle{P}$ at point $B$. Show that $\dfrac{1}{PA}+\dfrac{1}{PB}$ is a constant.

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Mass Points Exercises

Posted on July 22, 2021 by kevin

(AIME 2011) In triangle $ABC, AB=\dfrac{20}{11}AC.$ The angle bisector of $\angle A$ intersects $BC$ at point $D$, and point $M$ is the midpoint of $AD$. Let $P$ be the point of intersection of $AC$ and $BM$. Find $\dfrac{CP}{PA}$. SolutionWithout loss … Continue reading →

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Geometry Challenge – 2 ⭐⭐

Posted on July 22, 2021 by kevin

In acute $\triangle{ABC}$, $D$ and $E$ are on $AB$ and $AC$ respectively, and $CD\perp AB$, $BE\perp AC$. Draw perpendicular lines from $B$ and $C$ toward line $DE$, intersecting $DE$ at $F$ and $G$ respectively. Show that $DF=EG$.

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Geometry Challenge – 1 ⭐

Posted on July 20, 2021 by kevin

Let $ABCD$ be a quadrilateral such that $AB = AC, \angle BAC = 20^\circ, AD = CD,$ and $\angle ADC = 100^\circ$ . Show that $AB = BC + CD$.

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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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