# Circle of Apollonius

The Circle of Apollonius is another approach to defining a circle given by a line segment $AB$.

Theory

### TheCircleofApollonius

The circle of Apollonius is the locus of all points $P$ satisfying $\angle APB=v$ for a given angle $v$. Different values of $v$ result in different circles. The line segment $AB$ becomes a chord in this circle, and the angle $v$ becomes an inscribed angle spanning the circular arc $AB$.

The triangle $△AOB$ is an isosceles triangle, since the legs $AO$ and $BO$ are both equal to the radius of the circle. You therefore know that $\angle BAO$ and $\angle ABO$ are equal. You can now construct these at $A$ and $B$, when $v$ is given, with the help of the following formula:

 $\angle ABO=\angle BAO=90\text{°}-v$

The point where the angle rays meet each other is the center of a circle passing through the points $A$ and $B$. You can now use the radius and center to construct the circle with the use of a compass tool.

Example 1

Construct a triangle $\angle ABC$ where $AB=\text{}10\text{}\phantom{\rule{0.17em}{0ex}}\text{cm}$, angle $\angle ACB=\text{}45\text{}\text{°}$, the distance from $C$ down to the line segment $AB$ is $\text{}5\text{}\phantom{\rule{0.17em}{0ex}}\text{cm}$ and $C$ is closer to $A$ than $B$.

Draw an auxiliary figure:

You start by constructing the line $AB=10\phantom{\rule{0.17em}{0ex}}\text{cm}$. Then you construct perpendicular lines from both $A$ and $B$ and set the height to $5$ cm.

You must now find a way to construct $\angle C$, and it’s here you use the circle of Apollonius. The only thing you know is that $\angle C$ is supposed to be on the dotted line that is $5$ cm from $AB$, but you don’t know where.

On the other hand, you do know from the circle of Apollonius that if you make a circle that goes through $A$ and $B$, the center $S$ of the circle forms the central angle to any inscribed angle that spans $AB$. Therefore you can use that $\angle C=45\text{°}$ to find the angles $\angle ABS$ and $\angle BAS$. You can use Formula ():

 $\angle ABS=\angle BAS=90\text{°}-45\text{°}=45\text{°}$

Thus, you construct $45$° angles at $A$ and $B$ so that they turn inwards. The intersection point $S$ of these angle rays is the center of the circle of Apollonius.

Since $\angle BAS$ and $\angle SBO$ are both $45$°, the angle $\angle S=90\text{°}$.

This fits with $\angle S$ being the central angle of $\angle C$. Thus, you can draw the circle with center in $S$ and radius $SA$. You find $\angle C$ where the circle crosses the dotted line. This happens at two points, but the task tells you to choose the one closest to $A$. Thus, you can draw the lines between $A$, $B$ and $C$ so that the triangle is complete.

Isn’t it beautiful?