Circle of Apollonius

Draw an auxiliary figure:

You start by constructing the line $AB=10\text{cm}$. 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 ():

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?