Problem 229

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Problem 229: (2011 AMC 12a, 25)

Triangle $ABC$ has $\angle BAC = 60^{\circ}$, $\angle CBA \leq 90^{\circ}$, $BC=1$, and $AC \geq AB$. Let $H$, $I$, and $O$ be the orthocenter, incenter, and circumcenter of $\triangle ABC$, respectively. Assume that the area of pentagon $BCOIH$ is the maximum possible. What is $\angle CBA$? $\textbf{(A)}\ 60^{\circ} \qquad$ $ \textbf{(B)}\ 72^{\circ} \qquad$ $ \textbf{(C)}\ 75^{\circ} \qquad$ $ \textbf{(D)}\ 80^{\circ} \qquad$ $ \textbf{(E)}\ 90^{\circ}$

Solution

Exercise 229: Solution

Triangle $ABC$ has $\angle BAC=60^\circ$, $\angle CBA\le90^\circ$, $BC=1$, and $AC\ge AB$. Let $H$, $I$, and $O$ be the orthocenter, incenter, and circumcenter of $\triangle ABC$, respectively. Assume that the area of pentagon $BCOIH$ is the maximum possible. What is $\angle BCA$?

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