Inequality with Circle and Triangle in Euclidean Geometry

AI Thread Summary
The discussion centers on a geometry problem involving a circle and a triangle, where the user is struggling with two specific parts of the solution. Key points include a query about the validity of a geometric implication that lacks clear explanation or proof. The user references Euclid's propositions to support their reasoning about angle relationships and the implications for side lengths. They argue that understanding one angle's relationship to another leads to a conclusion about the lengths of the sides of the triangle. The conversation highlights the need for clearer explanations in geometric proofs.
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Homework Statement



Please see below...

Homework Equations



Please see below...

The Attempt at a Solution



Hi. This question is on geometry with circle and triangle. I am stuck only on 2 parts of the solution and not the whole solution...

Thank you...

http://img256.imageshack.us/img256/9475/gtewp249no24.jpg

[ How did they get this? They never explained or proved this and it is NOT obvious from the picture.

Blue: How is that implication true? By what theorem or reasoning?
 
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To see that the red statement is true, observe that \angleADO = \angleCDB > \angleBOD by Euclid I.32 (book I proposition 32).

In fact, that statement is superfluous. Once you know that \angleADO > \angleBDO = \angleADC = \angleAOD + \angleOAD, you know that \angleADO > \angleOAD, and by Euclid I.19, OA > OD.
 

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