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 Apollonius' Tangency Problem For Three Circles. Level: High School, SAT Prep, College geometry

Given three fixed circles, find a circle tangent to all three. The eight solutions are illustrated below. Click the Next button below  to browse through the 10 steps.

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Instructions Next screen: Next button or Right arrow, Previous screen: Previous button or Left arrow, First screen: Up arrow, Last screen: Down Arrow

Apollonius of Perga 262BC - 190BC, Greek mathematician of the Alexandrian school, made historical contributions to mathematics, astronomy and ballistics. His famous book Conics introduced the terms parabola, ellipse and hyperbola. According to Pappus, the Apollonian problem was included in his treatise De Tactionibus ("About Tangencies").

 

I have used Geometry Expressions to visualize these geometric forms and check out a variety of conjectures. Geometry Expressions is the world's first Interactive Symbolic Geometry System. This means: Geometric figures can be defined by either Symbolic Constraints or numeric locations.

You can download Geometry Expressions 1.0 Free Trial. It is a fully featured 30 day evaluation copy of the software. All constraints, constructions and measurements are available for you to use.
 

 

See also:

 

1.

Eyeball Theorem: Animated Angle to Geometry Study.

2.

Seven Circles Theorem

3.

Equal Incircles Theorem

4.

Archimedes' Book of Lemmas
 

5.

Butterfly Theorem

6.

Triangle Centers

7.

Euler and his beautiful and extraordinary formula

8.

Semiperimeter and incircle

9.

Semiperimeter and excircles of a triangle

10.

Semiperimeter, incircle and excircles of a triangle

11.

Nagel Point Theorem

12.

Gergonne Point Theorem

Gergonne Point Theorem

13.

Monge & d'Alembert Three Circles Theorem I with Dynamic Geometry

14.

Monge & d'Alembert Three Circles Theorem II with Dynamic Geometry

15.

Miquel's Pentagram with Dynamic Geometry

16.

Miquel's Pentagram Theorem

Miquel's Pentagram Theorem. Proof

17.

Johnson's Theorem

18.

Simson Line

19.

Angle between two Simson Lines

20.

Sangaku

Sangaku Problem

21.

Newton's Theorem

22.

The Bevan Point

23.

Clifford's Circle Chain Theorems

24.

Kurschak's Tile and Theorem

25.

Animated Angle to Geometry Study

26.

Proposed Problem 28
Right Triangle, altitude, incircles and inradius.

27.

Steiner Point
 

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Last updated: March 1, 2008