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Subject: "Monge via Desargues"

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Conferences The CTK Exchange High school Topic #383
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Reading Topic #383

Don Johnson

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Apr-07-09, 09:12 PM (EST)

"Monge via Desargues"

My Math teacher was challenging me to prove how the tangents formed by the circles are collinear and when I showed him the proof off of cut-the-knot( https://www.cut-the-knot.org/Curriculum/Geometry/MongeDesargues.shtml ) he asked me why exactly are the lines O1A1||O2A2||O3A3, but I had no idea. He wants to know which theorem or proof states how they are parallel. No need to rush and thanks in advance for the help.

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alexb
Charter Member
2352 posts

Apr-07-09, 09:23 PM (EST)

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2. "RE: Monge via Desargues"
In response to message #0

Circles (O₁) and (O₂) are homothetic with center of homothety at, say, P. The points A₁ and A₂ correspond to each other and so are the centers O₁ and O₂. This means that

the points P, A₁, A₂ are collinear,
the points P, O₁, O₂ are also collinear, and
PA₁ / PA₂ = PO₁ / PO₂,

which implies that A₁O₁ || A₂O₂.

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