Circle Geo Query (2 Viewers)

[apollo1]

What is the theorem that relates angles in same segment to cyclic quads called?

I recall vaguely that if angles standing on the same arc are equal then all four points of contact with circle are concyclic. but i dont no how to describe it as a theorem.

 

[thorax94]

if two congruent angles with their vertices in the same half-plane subtend the same segment on the boundary of that half-plane, then the endpoints of that segment together with the vertices of the angles form a cyclic quadrilateral. Something like that?

 

[Drongoski]

I think the same result can be viewed in 3 ways:

a) angles subtended at the circumference of a circle from a (same) chord are equal

b) angles subtended at the circumference of a circle from a (same) arc are equal

c) angles in any given segment of a circle are equal

All above are equivalent - just 3 different ways af seeing the same thing. The last one can be cited most briefly as: "angles in a segment are equal"

Edit

I misaddressed the query. Thorax gave the right one.

We of course prefer to cite the rule as briefly and clearly as we can. Here's one suggestion.

If two angles subtended from a line segment formed by 2 vertices(or points) at the other 2 vertices, on the same side of the line segment, are equal, then the 4 vertices are concyclic.

To say 4 points are concyclic is equivalent to saying the quadrilateral formed by them is cyclic.

 

[apollo1]

I think the same result can be viewed in 3 ways:

a) angles subtended at the circumference of a circle from a (same) chord are equal

b) angles subtended at the circumference of a circle from a (same) arc are equal

c) angles in any given segment of a circle are equal

All above are equivalent - just 3 different ways af seeing the same thing. The last one can be cited most briefly as: "angles in a segment are equal"

im not talking about those ones. hold on ill provide an example.

 

[apollo1]

in this diagram assume that BC is a chord. im trying to say that since angle BAC = angle BDC and they are standing on same chord the four points must be concyclic and hence they form a cyclic quad.

im trying to find the theorem for this.

 

[apollo1]

If two angles subtended from a segment formed by 2 vertices(or points) at the other 2 vertices are equal, then the 4 vertices are concyclic.

yes thts the one i was looking for.

 

[apollo1]

So "IF line segment joining two points subtends equal angles at two other points, then these four points are concyclic"

can i use this to prove cyclic quads? or are there any limitations?

 

[Drongoski]

yes thts the one i was looking for.

I think you must also qualify that with "on the same side" to be correct; I amended my post.

 

[apollo1]

I think you must also qualify that with "on the same side" to be correct; I amended my post.

wat do u mean by on the same side?

 

[Drongoski]

So "IF line segment joining two points subtends equal angles at two other points, then these four points are concyclic"

can i use this to prove cyclic quads? or are there any limitations?

The 2 angles must be on the same side of the line segment joining the 2 earlier points.

 

[apollo1]

The 2 angles must be on the same side of the line segment joining the 2 earlier points.

oh i got it. tnk u very much

 

[Drongoski]

Say you have 2 points A and B; join them to form a line segment AB. Say point C is above AB and point D is below AB and angle ACB = angle ADB. You cannot in this case conclude A, B, C and D are concyclic.

 

[Drongoski]

apollo1 - are you doing yr 11 atm?

 

[apollo1]

apollo1 - are you doing yr 11 atm?

no im in year 12 doing 4U.

 

[Drongoski]

None too soon then.

 

[apollo1]

None too soon then.

wat do u mean?

 

[Drongoski]

With exam only a few weeks away you haven't much time left - so none too soon. Maybe I'm using archaic English ha ha.

 

[Alkanes]

I hate circle geometry T_T...

 

[apollo1]

I hate circle geometry T_T...

random insight. but srsly learn to like it bcuz it is a topic regularly used by teachers to distinguish between top students.
in my half yearly for 3U the whole last page was hard circle geo questions. and the top 3 marks were like 5-7 marks above the rest.

 

[Alkanes]

Hence why i hate it.