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**Double tangent circles and focal properties of sphero-conics.**
*(English)*
Zbl 1177.51016

Let \(S^2\) be the unit sphere of the real Euclidean \(3\)-space. A sphero-conic is the intersection of \(S^2\) by a quadratic cone whose vertex coincides with the center of \(S^2\).

The author gives two proofs for each of the following two theorems: 5mm

Four excellent figures accompany the considerations of the article.

The author gives two proofs for each of the following two theorems: 5mm

- 1.
- The locus of all points on \(S^2\) such that the absolute sum or difference of tangent distances to two fixed circles \(K_1\), \(K_2\) is constant is a sphero-conic \(C\) with double tangent circles \(K_1\), \(K_2\); the centers of \(K_1\) and \(K_2\) lie on the same axis of \(C\).
- 2.
- If \(C\) is a sphero-conic on \(S^2\) and \(K_1,K_2\subset S^2\) are double tangent circles to \(C\) with centers on the same axis of \(C\), then the absolute sum or difference of tangent distances from points of \(C\) to \(K_1\) or \(K_2\) is constant.

Four excellent figures accompany the considerations of the article.

Reviewer: Rolf Riesinger (Wien)

### MSC:

51M09 | Elementary problems in hyperbolic and elliptic geometries |

51N05 | Descriptive geometry |

51N25 | Analytic geometry with other transformation groups |

51M04 | Elementary problems in Euclidean geometries |