2007年10月18日 星期四

圓周角是圓心角的兩倍的逆定理:解答

本文是《圓周角是圓心角的兩倍……的逆定理?》一文的續集。

「圓周角是圓心角的兩倍」這定理是可逆的,以下是其中一個逆定理的版本:

設 O 為某圓的圓心,B、C 為圓上的兩點。若 ∠BOC = 2∠BAC,則 A 也位於此圓的圓周上。(後記:此版本有誤,見文章的留言部分。)

證明這逆定理很簡單,只需用到原定理和「同弓形的圓周角(angles in the same segment)」定理即可,讀者不妨試試。

以上的版本是先固定圓心角,然後指出凡是圓心角的一半的皆是圓周角。我們可否先固定圓周角呢?考慮以下版本:

設 A、B、C 為某圓上的三點。若 ∠BOC = 2∠BAC,則 O 是這個圓的圓心。

很可惜,以上命題是錯的(只需考慮滿足 ∠BOC = 2∠BAC 的點 O 的軌跡)。要一個先固定圓周角的逆定理版本的話,就得加入一個條件:

設 A、B、C 為某圓上的三點。若 O 位於 BC 的垂直平分線上,且 ∠BOC = 2∠BAC,則 O 是這個圓的圓心。(後記:此版本有誤,見文章的留言部分。)

4 則留言:

Angela 提到...

For the first "converse", point A could lie on any circle whose radius is the same as the circle with O as centre and passes through B and C.

Similarly, for the second "converse", point O could be the centre of another circle as well... I think locus of O is a circle in the plane perpandicular to the plane of the original circle and contains all the perpandicular bisectors of BC.

Is there a more accurate version of converse of 'angle at centre is double of angle at circumference'?

Kahoo 提到...

Oh, you are right. Well, first of all, we shall assume that all points are on the same plane, a "normal assumption" when we are doing plane geometry. Under this assumption, there are actually exactly two circles for both versions, being symmetric about BC. In order that the converses hold, we need A and O to be on the same side of BC. To be precise, the two versions should be as follows:

Version 1
Let O be the centre of a circle and B, C be two points on the circumference. If A lies on the same side of BC as O and ∠BOC = 2∠BAC, then A also lies on the circumference of the circle.

Version 2
Let A, B, C be three points on a circle. If ∠BOC = 2∠BAC, O lies on the perpendicular bisector of BC and O lies on the same side of BC as A, then O is the centre of the circle.

Version 2 doesn't very much look like a "converse". :P

Angela 提到...

What if ∠BAC is obtuse?

Kahoo 提到...

Oh, I overlooked that again. There are two ways to fix the problem. (Only Version 2 needs revision because in Version 1, ∠BOC is assumed to mean the non-reflex angle.) One is to further digress into the cases where ∠BAC is acute or obtuse, but that will further complicate the "converse". The other is to add the restriction that ABC (in this order) is a minor arc of the circle, discarding the case when ∠BAC is obtuse.