Geometry: Similarity and Power of a Point

[wubie]

Hello,

I am not sure how to interpret this question:

Lines AB and CD intersect at a point P, where P is not between A and B. Show that if (PA)(PB) = (PC)(PD), then ABCD is a cyclic quadrilateral. [Note: The notation (XY) is used to represent the directed distance from X to Y.]

Now should I assume that because it WAS explicitly stated that P is not between A and B and that it WAS NOT explicitly stated P is not between C and D that in fact P IS between C and D? (Hope you can follow that).

So what do I have then? Lines AB and CD which intersect at P where P is not between A and B but P is between C and D? (Why don't they just draw a diagram? @#$!@#$!@#$%!#$!#$!). In fact, is it even possible to have a cyclic quad. if P is between CD and not between AB? I can only see a cyclic quad. happening if P is not between both AB and CD.



[?]


Opinions on the question would be appreciated. Thankyou.

 

[gnome]

My opinion is there's something wrong with the question.

If P is not between A and B, then isn't P at either A or B? And if so, don't you end up with a triangle rather than a quadrilateral?

 

[wubie]

The way I look at it is if the point of intersect is not between AB and not between CD then a simple convex cyclic quad. can be formed.

If the point of intersection is not between AB but the point of intersection is between CD then the resulting quadrilateral cannot be cyclic.

Since the question states if a condition is met a cyclic quad. results, this leads me to believe that point P must not be interior to AB or CD. But if that is the case, why did the question not specify the condition for both segments, AB and CD?

 

[Hurkyl]

Now should I assume that because it WAS explicitly stated that P is not between A and B and that it WAS NOT explicitly stated P is not between C and D that in fact P IS between C and D?

No. If "P is between C and D" was part of the hypothesis, they would have stated it. P is allowed to be anywhere WRT C and D.

However, you state that (XY) is the directed distance from X to Y. You can prove something about P's location WRT C and D using the directions of these segments.

Why don't they just draw a diagram?

Because pictures aren't acceptable hypotheses when working with full rigor. The problem with pictures is that they can often assume a tremendous amount of information without you noticing. There are lots of reasons why you can't, or won't want to draw a picture to describe a problem, so it's important to be able to understand things algebraically.

If P is not between A and B, then isn't P at either A or B? And if so, don't you end up with a triangle rather than a quadrilateral?

P can lie outside of the quadrilateral.

For instance, consider the trapezoid ABCD whose two bases are BC and DA. Draw the lines AB and CD and see where they intersect.

 

[wubie]

I see. I think I can see how a cyclic quad. can result from the above situtation then. The key is directed distances. But I am not sure if I am doing it correctly though.

Do I start with a point P and draw lengths PA, PB, PC, and PD from it? I don't remember how to use directed distances.

 

[HallsofIvy]

" I don't remember how to use directed distances."

Then you might want to start by looking up the definition of "directed distance".

 

[wubie]

Yes, I did. The thing is I haven't done geometry for over fifteen years. The prereq. for this course was just a first year math course - didn't matter what course as long as it was first year. This course started right in the middle text, chapter three. Directed distances were located in chapter one. So the class missed some basic geometry def. and theory (one of those things being directed distances).

Anyway, this assignment has already been submitted. And I wasn't able to do the question. But I still would like to know how to set up the question. I can do it once I get started, but I don't know how to set it up. I have looked at the def. of directed distances, but I didn' help me regarding this question.

Any explanations regarding the set up with this question would be appreciated.

Thankyou.

 

[Hurkyl]

Well, what is the definition of directed distance?


From it, you should be able to prove this lemma:

If A, B, and C are collinear, then (BA)(BC) is negative if and only if B is between A and C.

 

[wubie]

Def. given me:

Let points A and B be points on the line. The directed distance AB id defined as follows:

the directed distance AB is equal to

AB if A < B
0 if A = B
-AB if B < A

I will work on the lemma next. bbl.

 

[wubie]

I am not very good at proofs. But I will give this a shot.

If (BA)(BC) is negative then either:

1. (BA) is negative and (BC) is positive

or

2. (BA) is positive and (BC) is negative.


Assume BA is negative, then for the directed distance BA, -BA if A < B.

Assume BC is positive, then for the directed distance BC, BC if B < C.

Therefore A < B < C.

Conversely,

Assume BA is positive, then for the directed distance BA, BA if B < A.

Assume BC is negative, then for the directed distance BC, -BC if C < B.

Therefor C < B < A.

In both cases, B is between A and C.


That would be be my substandard proof.

okokokok. I think I see where you are leading me. If I know this then I know the orientation of (PA), (PB), (PC), and (PD).

In this case would A be between P and B and would C be between P and D then?

I would then have two lines that intersect at P. The points A,B,C,D would be points on a circle. P is a point outside of the circle. A,B,C,D would also be the vertices of a cyclic quad.

Is this a correct assessment?

 

[Hurkyl]

In this case would A be between P and B and would C be between P and D then?

No... all the lemma says for your problem is that if P is not between A and B, then P is not between C and D.

You'll have to apply one of those fun betweenness axioms if you need to pin P's location down more precisely. (In this case, if A is between P and B, then D will be between P and C)

 

[wubie]

Sorry. I wasn't thinking properly last night.

No... all the lemma says for your problem is that if P is not between A and B, then P is not between C and D.

Yes. I think I see now. From the question

P is not between A and B.

and

(PA)(PB) = (PC)(PD)[Note: The notation (XY) is used to represent the directed distance from X to Y.]

I can deduce that P is not between C and D either. Correct?

 

[Hurkyl]

Right.

 

[wubie]

Thanks for your help Hurkyl.