Category Archives: Geometry

$AB$ is the diameter of the circle center at $O$. $CD$ is tangent to the circle at $D$ and $AB\parallel CD$. $AC$ intersects the circle at $E$ and $AE=CE$. Find $\angle{ACD}$.

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.

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 $DF=EG$.

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