Given $\triangle ABC$ with $AB=BC$ and $\angle ABC=80^\circ$, and $O$ such that $\angle OAC=30^\circ$ and $\angle OCA=10^\circ$, find $\angle BOC$.

Question

I encountered the following problem in a math problem book, in a section, entitled Geometry.

In an $\Delta ABC$ with $AB=BC$ and $\angle ABC = 80^\circ$, the point $O$ is chosen so that $\angle OAC = 30^\circ$ and $\angle OCA = 10^\circ$. Find $\angle BOC$.

I struggled repeatedly trying to solve this using some reasonable selection of methods, and failed utterly. I can construct a solution using analytic geometry, but that is not a fair choice of methods for this section.

From simple geometry, I keep writing down systems of equations using various triangles in the picture, and always keep missing one constraint, as it seems there are infinite amount of solutions to the system, which I know is wrong since the analytic geometry approach yields a unique solution.

I am happy to add my infinite solution approach using the angle constraints, if you find it helpful. Please let me know if that's the case.

I would like to solve this using methods from geometry, without using more advanced techniques. Thank you very much, I really appreciate your help.

Answer 1

Extend $AO$ to meet $BC$ at $M$ . Note that $\widehat{AMB}=80^o$ (why?) and therefore $\widehat{MOC} = 40^o$ (why?).

Draw $BP$ , bisector of $\widehat B$ . Note that $\triangle ABC$ is symmetric with respect to $BP$ . Let $CN$ be the mirror image of $AM$ with respect to $BP$. Then $$\widehat{NCB} = \widehat{MAB} = 20^o$$ $$\widehat{NCO} = \widehat{NCA} - \widehat{OCA} = 30^o - 10^o = 20^o $$ Therefore we have: $$\triangle IOC = \triangle IBC$$ It follows that $BC = OC$, which means $\triangle COB$ is isosceles, hence: $$\widehat{BOC} = \widehat{OBC} = \frac12 (180 - \widehat{OCB}) = 70^o$$ Et voila!

Fun fact: the segments shown in cyan ($AM,CN,CO,AB,CB$) are of equal length.

Answer 2

Here is an "out of triangle thought" for this question. Law of sine can be applied twice to reveal a hidden equilateral triangle $\triangle BOC$ if O is allowed to be outside $\triangle ABC$. (There might be a simple geometric solution out there someone can find.)

Set BC=1

For $\triangle ABC$ , AC=$\frac{\sin 80^\circ}{\sin 50^\circ}$

For $\triangle AOC$, CO=$\frac{AC\sin 30^\circ}{\sin 140^\circ}=\frac{\sin 80^\circ\sin 30^\circ}{\sin40^\circ\sin 50^\circ}=\frac{\sin 80^\circ}{2\sin40^\circ\cos40^\circ}$=1

Therefore, $\triangle BOC$ is an isosceles triangle, with $\angle BCO=60^\circ$$\to$ $\angle BOC=60^\circ$

Answer 3

Let $P$ be the circumcentre of $\triangle AOC$. Extend $AO$ to meet $BC$ at $D$.

Observe that, $\angle APC=80^{\circ}$ and hence $\triangle BAC\cong \triangle PAC$ and thus $BC=PC$.

Also, notice that $\triangle POC$ is equilateral and thus $OC=PC$.

Coupled with the result obtained earlier, we get $BC=PC=OC$ and hence $\triangle BOC$ is isosceles.

Therefore, $\angle BOC=\frac {180-40}{2}=\boxed {70^{\circ}}$