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Some Remarks on de Longchamps Chain

Published online by Cambridge University Press:  20 January 2009

L. M. Brown
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The chain of circle theorems now associated with the name of de Longchamps may be stated as follows.

(i) Any two lines L (A), L (B) meet in a point (AB).

(ii) Any three lines L(A), L (B), L(C) determine three points (AB) (BC) (CA) which lie on a circle S (ABC) with centre (ABC),

(iii) Any four lines L(A), L(B), L(C), L(D) determine four circles S(ABC) S(ABD) S(ACD) S(BCD) which meet in a point P (ABCD) and four centres (ABC) (ABD) (ACD) (BCD) which lie on a circle S(ABCD) with centre (ABCD).

(iv) Any five lines L(A), L(B), L(C), L(D), L(E) determine five points like P (ABCD), which lie on a circle SM (ABCDE) (Miquel's Theorem), five circles like S (ABCD), which meet in a point P (ABCDE), and their five centres like (ABCD), which lie on a circle S (ABCDE) with centre (ABCDE).

(v) Any six lines L (A), … . , L (F) determine six circles like SM (ABCDE) which meet in a point PM (ABCDEF), six circles like S (ABCDE), which meet in a point P (ABCDEF) and six centres like (ABCDE) which lie on a circle S (ABCDEF) with centre (ABCDEF).

Type
Research Article
Information
Proceedings of the Edinburgh Mathematical Society , Volume 7 , Issue 3 , February 1946 , pp. 138 - 143
Copyright
Copyright © Edinburgh Mathematical Society 1946

References

REFERENCES

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