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 $DE$ ?

#Geometry

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.
Stay on topic — we're all here to learn more about math and science, not to hear about your favorite get-rich-quick scheme or current world events.

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

Math	Appears 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}`	$\boxed{123}$

Comments

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

A Former Brilliant Member - 4 years, 11 months ago

Can you give proof of your statement please

Sarthak Singla - 4 years, 11 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 $0<y<x<90^\circ$ , 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 $180^\circ > x+y > 90^\circ \iff 180^\circ > \angle BOA + \angle BOC >90^\circ$

A Former Brilliant Member - 4 years, 11 months ago

@A Former Brilliant Member – Was this helpful? @Sarthak Singla

A Former Brilliant Member - 4 years, 11 months ago

@A Former Brilliant Member – 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.

Sarthak Singla - 4 years, 11 months ago

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

Sarthak Singla - 4 years, 11 months ago

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

A Former Brilliant Member - 4 years, 11 months ago

Math	Appears 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}`	$\boxed{123}$