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File:Proof of Apollonius' definition of a circle.svg

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English: Illustration of a proof of Apollonius' definition of a circle.

Apollonius discovered that a circle could be defined as the set of points that have a given ratio of distances to two given points and , so .

First consider the point on the line segment between and , satisfying the ratio. By the definition and from the Angle bisector theorem the angles and are equal.

Next take the other point on the extended line , which satisfies the ratio. So

Also take some other point anywhere on the extended line . Also by the Angle bisector theorem the line bisects the exterior angle . Hence, and are equal and . Hence by Thales's theorem lies on the circle which has as a diameter.
Date
Source Own work
Author Salix alba

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Proof of Apollonius' definition of a circle

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28 January 2021

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Date/TimeThumbnailDimensionsUserComment
current13:39, 31 January 2021Thumbnail for version as of 13:39, 31 January 2021539 × 423 (21 KB)Salix albaremove highlighting on one vertex
13:37, 31 January 2021Thumbnail for version as of 13:37, 31 January 2021539 × 423 (22 KB)Salix albaMake sure AP it not tangent to circle
14:58, 28 January 2021Thumbnail for version as of 14:58, 28 January 2021535 × 391 (21 KB)Salix albaBetter scaling and cropping
14:33, 28 January 2021Thumbnail for version as of 14:33, 28 January 2021986 × 553 (21 KB)Salix albaUploaded own work with UploadWizard

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