Category Archives: Geometry

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 →

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 →

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 →