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Circle of Apollonius

The circle of apollonius

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

Theory

The Circle of Apollonius

The circle of Apollonius is the locus of all points P satisfying 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 BAO and ABO are equal. You can now construct these at A and B, when v is given, with the help of the following formula:

ABO = BAO = 90° 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 ABC where AB = 10cm, angle ACB = 45°, the distance from C down to the line segment AB is 5cm and C is closer to A than B.

Draw an auxiliary figure:

Auxiliary figure of triangle

You start by constructing the line AB = 10cm. 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 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 C = 45° to find the angles ABS and BAS. You can use Formula ():

ABS = BAS = 90° 45° = 45°

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.

Construction example using circle of apollonius 2

Since BAS and SBO are both 45°, the angle S = 90°.

This fits with S being the central angle of C. Thus, you can draw the circle with center in S and radius SA. 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

Construction example using circle of apollonius 3

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