Nice geometry problem

Let $ABC$ be a triangle. $D$ and $E$ lie on $AB$ such that $AD = AC, BE = BC$ and the points $D, A, B, E$ are collinear in that order. The bisectors of angle $A$ and $B$ intersect $BC, AC$ at $P$ and $Q$ respectively, and the circumcircle of $ABC$ at $M$ and $N$ respectively. Let $O_1$ be the circumcenter of $BME$ and $O_2$ be the circumcenter of $AND$. $AO_1$ and $BO_2$ intersect at $X$. Prove that $CX$ is perpendicular to $PQ$.

Source : Serbia $2008$ .

#Geometry

Easy Math Editor

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

b=c square

Sergio VIlla - 6 years, 8 months ago

Solution (SPOILER) :

We let $AB = c, BC = a, CA = b$ for simplicity.

Note that $AE \cdot AQ = (a + c) \cdot \frac{bc}{a + c} = bc$ , by Angle Bisector Theorem. Also, $AP \cdot AM = AP \cdot (AP + PM) = AP^2 + AP \cdot PM = AP^2 + BP \cdot CP$ , by Power of Point. Since by Stewart's Theorem, $AP^2 + BP \cdot CP = \frac{AB^2 \cdot CP + AC^2 \cdot BP}{BC} = \frac{c^2(\frac{ab}{b + c}) + b^2(\frac{ac}{b + c})}{a} = \frac{bc(b + c)}{b + c} = bc$ , we have $AP \cdot AM = bc$ .

Consider the following transformation (commonly known as $\sqrt{bc}$ -inversion.) :

1) Reflect every point $X$ across the angle bisector of $\angle BAC$ .

2) Invert about point $A$ with radius $\sqrt{bc}$ .

So, it follows immediately that $E$ and $Q$ map to each other. Similarly, $P$ and $M$ map to each other. Thus, the circumcircle of $BME$ is mapped to the circumcircle of $CPQ$ , with center $O$ . Therefore, $AO$ is isogonal to $AO_1$ in $\angle BAC$ , since the reflection of $O_1$ across the angle bisector of $\angle BAC$ is collinear with $A$ and $O$ . Similarly, $BO$ is isogonal to $BO_2$ in $\angle ABC$ . Thus, $O$ and $X$ are isogonal conjugates with respect to $\triangle ABC$ . Finally, since in triangle $CQP$ , $CX$ is isogonal with $CO$ , $C, H, X$ are collinear and thus $CX \perp PQ$ . ( $H$ is the orthocenter of $CPQ$ )

The last line follows from the fact that $O$ and $H$ are isogonal conjugates.

Zi Song Yeoh - 6 years, 8 months ago

An easy way to establish $AP\cdot AM=bc$ : we have $\angle BAM=\angle CAP$ (since $AP$ angle bisector) and $\angle ACB=\angle AMB$ since $ABMC$ cyclic. It follows that $\triangle ABM\sim \triangle APC$ so side ratios are equal.

Jubayer Nirjhor - 6 years, 8 months ago

Can you provide a diagramatical representation . I coudnt.draw a figure for the given information

Shehanaaz Sk - 6 years, 8 months ago

(a)=ad bxqpxmn-1-1X1x(a)(B)2x

Sergio VIlla - 6 years, 8 months ago

I m not able to figure out the diagram. Can someone help please?

Shreya Hardas - 6 years, 8 months ago

Who can prove 4=5

Atif Qureshi - 6 years, 8 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}`	$\boxed{123}$