# A geometry problem- I feel it is one

Hello! I have a doubt which came in my mind while solving a geometry question

Suppose we have 2 rays $OA$ and $OB$ with same end point $O$. Angle $AOB$ is any acute angle with value $x$.

another ray $OC$ is drawn such that it divides acute Angle $AOB$ into two parts, not necessarily equal.

Suppose there is a point $D$ on the ray $OA$.

My question is that, will a point $E$ always exist on ray $OB$ such that $OC$ bisect $DE$?

Note by Sarthak Singla
3 years, 8 months ago

This discussion board is a place to discuss our Daily Challenges and the math and science related to those challenges. Explanations are more than just a solution — they should explain the steps and thinking strategies that you used to obtain the solution. Comments should further the discussion of math and science.

When posting on Brilliant:

• Use the emojis to react to an explanation, whether you're congratulating a job well done , or just really confused .
• Ask specific questions about the challenge or the steps in somebody's explanation. Well-posed questions can add a lot to the discussion, but posting "I don't understand!" doesn't help anyone.
• Try to contribute something new to the discussion, whether it is an extension, generalization or other idea related to the challenge.

MarkdownAppears as
*italics* or _italics_ italics
**bold** or __bold__ bold
- bulleted- list
• bulleted
• list
1. numbered2. list
1. numbered
2. list
Note: you must add a full line of space before and after lists for them to show up correctly
paragraph 1paragraph 2

paragraph 1

paragraph 2

[example link](https://brilliant.org)example link
> This is a quote
This is a quote
    # I indented these lines
# 4 spaces, and now they show
# up as a code block.

print "hello world"
# I indented these lines
# 4 spaces, and now they show
# up as a code block.

print "hello world"
MathAppears as
Remember to wrap math in $$ ... $$ or $ ... $ to ensure proper formatting.
2 \times 3 $2 \times 3$
2^{34} $2^{34}$
a_{i-1} $a_{i-1}$
\frac{2}{3} $\frac{2}{3}$
\sqrt{2} $\sqrt{2}$
\sum_{i=1}^3 $\sum_{i=1}^3$
\sin \theta $\sin \theta$
\boxed{123} $\boxed{123}$

Sort by:

I think E would also exist if angleBOA+angleBOC=90

- 3 years, 8 months ago

It would be degenerate as $E=O$ in that case.

- 3 years, 8 months ago

There will exist such a point $E$ only if $\angle BOA + \angle BOC$ is obtuse.

- 3 years, 8 months ago

- 3 years, 8 months ago

My proof is via complex numbers.

Without loss of generality assume that $OB$ represents the real axis.

Let $D=\lambda e^{ix} ; \; \lambda \in \mathbb{R}^+$.

Let $D' (=\lambda' ; \; \lambda' \in \mathbb{R}^+)$ be any point on $OB$.

Now, $D'=E$ if $\text{Arg} \left( \frac{D+D'}{2} \right) = y$ where $y = \angle BOC$

Simplifying the above condition, we get $\frac{\lambda'}{\lambda} = \sin x ( \tan y - \cot x)$

As the LHS is positive, so is the RHS.

Therefore,

$\tan y > \cot x$

Now, as $0, we get (by using the cosine - sum and difference formulas)

$0>\cos (x+y)$

Due to the constraints on the angles $x$ and $y$, this again simplifies to $180^\circ > x+y > 90^\circ \iff 180^\circ > \angle BOA + \angle BOC >90^\circ$

- 3 years, 8 months ago

- 3 years, 8 months ago

Thanks for the solution. Can it not be proved by some simpler method? I feel it would take me quite a few years to decipher this solution as I am unaware of complex numbers.

- 3 years, 8 months ago