Author Archives: kevin

Geometry – Pythagoras’ Theorem

Posted on May 6, 2022 by kevin

In $\triangle{ABC}$, $AM$ is the median on the side $BC$. Prove that $AB^2+AC^2=2(AM^2 + BM^2)$ For $\triangle{ABC}$, $O$ is an inner point, and $D$, $E$, $F$ are on $BC$, $CA$, $AB$ respectively, such that $OD\perp BC$, $OE\perp CA$, and $OF\perp … Continue reading →

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Geometry – Sides and Angles of Triangles

Posted on April 22, 2022 by kevin

As shown in the diagram below, in $\triangle{ABC}$, $\angle{B}\gt \angle{C}$, $AD$ is the bisector of the $\angle{BAC}$, $AE\perp BC$ at $E$. Prove that $\angle{DAE}=\dfrac{1}{2}(\angle{B}−\angle{C})$. 2. There are four points $A$, $B$, $C$, $D$ on the plane, such that any three … Continue reading →

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Geometry Challenge – 12

Posted on April 4, 2022 by kevin

Acute $\triangle{ABC}$ is inscribed inside circle centered at $O$. $P$ is on $BC$ and $AP\perp BC$, and $\angle{ACB}>\angle{ABC}$. Prove the following: $\angle{BAC}+\angle{OBC}=90^\circ$ $\angle{OAP}=\angle{ACB}-\angle{ABC}$ If $\angle{ACB}-\angle{ABC}\ge 30^\circ$, and $MB=MC$, then $MP\gt CP$ If $\angle{ACB}-\angle{ABC}\ge 30^\circ$, then $\angle{BAC}+\angle{POC}<90^\circ$ Click here for the … Continue reading →

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Geometry Challenge – 11

Posted on March 21, 2022 by kevin

Let $D$ be an arbitrary point on the side $BC$ of a given triangle $ABC$ and let $E$ be the intersection of $AD$ and the second external common tangent of the incircles of triangles $ABD$ and ACD. As $D$ assumes … Continue reading →

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Geometry Challenge – 10

Posted on March 14, 2022 by kevin

In the diagram below, two congruent semi-circles that are tangent to each other, are inscribed inside a bigger semi-circle with its diameter as 10 unit. The area of the shaded region can be expressed as $\dfrac{a}{b}\pi$, where $a$ and $b$ … Continue reading →

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