Circle of Apollonius

The circle of apollonius

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

Theory

The Circle of Apollonius

The circle of Apollonius is the locus of all points

P

satisfying

∠ A P B = v

for a given angle

v

. Different values of

v

result in different circles. The line segment

A B

becomes a chord in this circle, and the angle

v

becomes an inscribed angle spanning the circular arc

A B

The triangle

△ A O B

is an isosceles triangle, since the legs

A O

and

B O

are both equal to the radius of the circle. You therefore know that

∠ B A O

and

∠ A B O

are equal. You can now construct these at

A

and

B

, when

v

is given, with the help of the following formula:

∠ A B O = ∠ B A O = 90^{\circ} - 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 $∠ A B C$ where $A B = 10 cm$ , angle $∠ A C B = 45^{\circ}$ , the distance from $C$ down to the line segment $A B$ is $5 cm$ and $C$ is closer to $A$ than $B$ .

Draw an auxiliary figure:

Auxiliary figure of triangle

You start by constructing the line $A B = 10 cm$ . Then you construct perpendicular lines from both $A$ and $B$ and set the height to $5$ cm.

Construction example using circle of apollonius 1

You must now find a way to construct $∠ C$ , and it’s here you use the circle of Apollonius. The only thing you know is that $∠ C$ is supposed to be on the dotted line that is $5$ cm from $A B$ , 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 $A B$ . Therefore you can use that $∠ C = 45^{\circ}$ to find the angles $∠ A B S$ and $∠ B A S$ . You can use Formula (0.0):

∠ A B S = ∠ B A S = 90^{\circ} - 45^{\circ} = 45^{\circ}

Thus, you construct $45$ $^{\circ}$ 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.

Construction example using circle of apollonius 2

Since $∠ B A S$ and $∠ S B O$ are both $45$ $^{\circ}$ , the angle $∠ S = 90^{\circ}$ . This fits with $∠ S$ being the central angle of $∠ C$ . Thus, you can draw the circle with center in $S$ and radius $S A$ . You find $∠ C$ 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?

Construction example using circle of apollonius 3

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