## Construct a line through A

For discussing Olympiad level Geometry Problems
Enthurelxyz
Posts: 17
Joined: Sat Dec 05, 2020 10:45 pm
Contact:

### Construct a line through A

Let $A$ be one of the common points of two intersecting circles. Through $A$ construct a line on which the two circles cut out equal chords.
and miles to go before we sleep
and miles to go before we sleep

Anindya Biswas
Posts: 200
Joined: Fri Oct 02, 2020 8:51 pm
Contact:

### Solution :

Let's name the circles $\Gamma_1$ and $\Gamma_2$. Let's assume $\Gamma_1\cap\Gamma_2=\{O,A\}$. Let's construct line $PQ$ such that $A\in PQ, PQ\cap\Gamma_1=P, PQ\cap\Gamma_2=Q$. Let $M$ be the midpoint of the segment $PQ$. Let's construct line $l$ through $O$ such that $\measuredangle (OP, l)=\measuredangle MOA$. Let $X=l\cap\Gamma_1$. The line $XA$ is the line on which $\Gamma_1$ and $\Gamma_2$ cut out equal chords.

Proof :
Let $Y=\Gamma_2\cap XA$. By Spiral Similarity, $OXAY\sim OPMQ$. Since $M$ is the midpoint of segment $PQ$, $A$ must be the midpoint of segment $XY$. Which completes our proof.

Another approach using Homothety :
Note that we could also solve it in this way,
Let's construct a circle $\Omega$ such that $\text{Radius}(\Gamma_1)=\text{Radius}(\Omega)$, $\Omega$ is externally tangent to $\Gamma_1$ at $A$. Let $Y=\Gamma_2\cap\Omega$. Then $YA$ is our wanted line.
"If people do not believe that mathematics is simple, it is only because they do not realize how complicated life is."
John von Neumann