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Quanta magazine https://www.quantamagazine.org/new-math-revives-geometrys-oldest-problems-20250926/ just reported the following:

In the third century BCE, Apollonius of Perga asked how many circles one could draw that would touch three given circles at exactly one point each. It would take 1,800 years to prove the answer: eight.

If you believe Quanta magazine that this question was indeed asked 1800 years ago and solved only now, then:

(a) This problem was open at the time this Mathoverflow question was asked

(b) It was open for about 1800 years, while other answers talk about problems about 200 years old, and

(c) It is a geometry problem. It is, like many other geometry problems, can be written in coordinates and reduced to some equations, but this is just one of several possible solution tools - by its nature it is still a geometry problem!

Quanta magazine https://www.quantamagazine.org/new-math-revives-geometrys-oldest-problems-20250926/ just reported the following:

In the third century BCE, Apollonius of Perga asked how many circles one could draw that would touch three given circles at exactly one point each. It would take 1,800 years to prove the answer: eight.

(Image from Quanta article.)

If you believe Quanta magazine that this question was indeed asked 1800 years ago and solved only now, then:

(a) This problem was open at the time this Mathoverflow question was asked

(b) It was open for about 1800 years, while other answers talk about problems about 200 years old, and

(c) It is a geometry problem. It is, like many other geometry problems, can be written in coordinates and reduced to some equations, but this is just one of several possible solution tools - by its nature it is still a geometry problem!

Quanta magazine https://www.quantamagazine.org/new-math-revives-geometrys-oldest-problems-20250926/ just reported the following:

In the third century BCE, Apollonius of Perga asked how many circles one could draw that would touch three given circles at exactly one point each. It would take 1,800 years to prove the answer: eight.

If you believe Quanta magazine that this question was indeed asked 1800 years ago and solved only now, then:

(a) This problem was open at the time this Mathoverflow question was asked

(b) It was open for about 1800 years, while other answers talk about problems about 200 years old, and

(c) It is a geometry problem. It is, like many other geometry problems, can be written in coordinates and reduced to some equations, but this is just one of several possible solution tools - by its nature it is still a geometry problem!

Quanta magazine https://www.quantamagazine.org/new-math-revives-geometrys-oldest-problems-20250926/ just reported the following:

In the third century BCE, Apollonius of Perga asked how many circles one could draw that would touch three given circles at exactly one point each. It would take 1,800 years to prove the answer: eight.

If you believe Quanta magazine that this question was indeed asked 1800 years ago and solved only now, then:

(a) This problem was open at the time this Mathoverflow question was asked

(b) It was open for about 1800 years, while other answers talk about problems about 200 years old, and

(c) It is a geometry problem. It is, like many other geometry problems, can be written in coordinates and reduced to some equations, but this is just one of several possible solution tools - by its nature it is still a geometry problem!

Quanta magazine https://www.quantamagazine.org/new-math-revives-geometrys-oldest-problems-20250926/ just reported the following:

In the third century BCE, Apollonius of Perga asked how many circles one could draw that would touch three given circles at exactly one point each. It would take 1,800 years to prove the answer: eight.

If you believe Quanta magazine that this question was indeed asked 1800 years ago and solved only now, then:

(a) This problem was open at the time this Mathoverflow question was asked

(b) It was open for about 1800 years, while other answers talk about problems about 200 years old, and

(c) It is a geometry problem. It is, like many other geometry problems, can be written in coordinates and reduced to some equations, but this is just one of several possible solution tools - by its nature it is still a geometry problem!

Quanta magazine https://www.quantamagazine.org/new-math-revives-geometrys-oldest-problems-20250926/ just reported the following:

In the third century BCE, Apollonius of Perga asked how many circles one could draw that would touch three given circles at exactly one point each. It would take 1,800 years to prove the answer: eight.

If you believe Quanta magazine that this question was indeed asked 1800 years ago and solved only now, then:

(a) This problem was open at the time this Mathoverflow question was asked

(b) It was open for about 1800 years, while other answers talk about problems about 200 years old, and

(c) It is a geometry problem. It is, like many other geometry problems, can be written in coordinates and reduced to some equations, but this is just one of several possible solution tools - by its nature it is still a geometry problem!