09/05/2025
$D$ is on side $BC$ of $\triangle{ABC}$ such that $BD=2$, $CD=1$, $\angle{ADB}=60^{\circ}$, $\angle{ACB}=30^{\circ}$, find $\angle{ABC}$.
Solve:
\(\begin{multline}
\shoveleft \text{Let } E \text{ on }AD \text{ such that }CD=DE \\
\shoveleft \implies BE=CE=\sqrt{3}, BE\perp DE\\
\shoveleft \implies \angle{CBE}=\angle{BCE}=30^{\circ}=\angle{CED}\\
\shoveleft \implies \angle{ACE}=45^{\circ}-30^{\circ}=15^{\circ}\\
\shoveleft \implies \angle{ACE}=15^{\circ}\implies CE=AE=BE\\
\shoveleft \implies \angle{ABE}=\angle{BAE}=45^{\circ}\\
\shoveleft \implies \angle{ABC}=\bbox[5px, border: 1px solid black]{75^{\circ}}
\end{multline}\)
09/10/2025
In $\triangle{ABC}$, $CD\perp AB$, $\angle{BCD}=30^{\circ}$, $E$ is on $CD$ such that $\angle{CAE}=\angle{DAE}=10^{\circ}$, find $\angle{CBE}$.
Solve:
\(\begin{multline}
\shoveleft \text{Let }E' \text{ be the reflection of }E \text{ along }AB \implies \angle{E'AD}=\angle{DAE}=10^{\circ}\\
\shoveleft \text{Let }F \text{ be the circumcenter of }\triangle{CAE'} \implies \angle{CFE'}=2\angle{CAE'}=60^{\circ} \\
\shoveleft \implies \triangle{CE'F} \text{ is equilateral }\implies AF=CF=E'F=CE'\\
\shoveleft \angle{CFE'}=60^{\circ} \implies \angle{FE'A}=20^{\circ}=\angle{FAE'} \implies F \text{ is on }AE\\
\shoveleft \text{Let }E'F\cap AD=G, \angle{E'GD}=30^{\circ}=\angle{EGD}=\angle{BCD} \\
\shoveleft \implies BCGE' \text{ is cyclic}, \angle{EGE'}=60^{\circ}=\angle{E'CF}\implies ECFG\text{ is cyclic}\\
\shoveleft \implies \angle{EFG}=40^{\circ}=\angle{ECG}=\angle{E'BG}=\angle{EBG}\\
\shoveleft \implies \angle{CBE}=\angle{CBD}-\angle{EBD}=60^{\circ}-40^{\circ}=\bbox[5px, border: 1px solid black]{20^{\circ}}\\
\end{multline}\)