Geometry Proof: Tips & Theorems to Solve It

In summary, the conversation discusses a geometry problem involving circles with a given radius and center point, and the points p, p', and q. The circle at the origin, though seemingly redundant, plays a crucial role in solving the problem. The solution involves using analytic geometry and calculus, and the significance of the circle at the origin is that it helps prove the relationship between the angles subtended by PP' and qq' at the origin and center of the first circle. By constructing two triangles and proving the included angles to be congruent, the solution can be proven to be 2 units.
  • #1
SpartaBagelz
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Mod note: Member warned that homework questions must be posted in the Homework & Coursework sections
http://imgur.com/zGB2dnY

Was given this problem a few weeks ago and I'm not sure how I should be approaching it. Please let me know which theorems I should look into in order to solve the problem.
 
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  • #2
As I can see the the circle with radius 1 and centered on origin seems tobe redundant. Then why is it given. may be that is some hint. I am also trying to solve. for p(√2,0) the required distance can be easily proved = 2. Tr it then try to solve the general problem.
 
  • #3
Let'sthink said:
As I can see the the circle with radius 1 and centered on origin seems tobe redundant. Then why is it given. may be that is some hint. I am also trying to solve. for p(√2,0) the required distance can be easily proved = 2. Tr it then try to solve the general problem.

Ok I figured it out but the middle circle actually does play a crucial role. So first the center of the left circle to the radius at the right is sqrt(2). Subtract 1 to find the space in between the origin and that spot on the right. If we set that point on the right of the left circle to be point q we can use it. From q to the center of the right has to be .59 because you subtract .41 from 1. Then if we set b to be (sqrt2,0) we know from the right center to the point is sqrt2. Add the .59 to it and you get two.
 
  • #4
I think you are referring to the situation when p and p' coincide or respectively coincide with intersection points of the circles. For those special cases we can easily prove that the distance is 2 and for the latter case the extra circle helps. But the general problem is still unsolved for me.
 
  • #5
SpartaBagelz said:
http://imgur.com/zGB2dnY

Was given this problem a few weeks ago and I'm not sure how I should be approaching it. Please let me know which theorems I should look into in order to solve the problem.
Drawing is not at scale.
 
  • #6
theBin said:
Drawing is not at scale.
It looks reasonably close to scale to me.
upload_2016-5-5_14-44-32.png
 
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  • #7
SammyS said:
It looks reasonably close to scale to me.
View attachment 100324
Sorry, now I see that it is at scale. I guess it could be proven by analytic geometry combined with (infinitesimal) calculus. High school geometry in the plan, permits only to make a series of tests for different positions of the points p, p' and q;
 
  • #8
I think it has to do with construction of two triangles with sides √2 and 1 but the angle is not included one.
 
  • #9
I have understood the significance of the circle at the origin of radius 1. That immediately tells us that the angle which PP' subtends at origin is twice the angle which qq' subtends at center (-1, 0) of the first circle. This can help us to prove the result.
 
  • #10
Let point (-1. 0) be labelled as a and (1, 0) as b and (0, 0) as o. Consider the triangles: qbp and qba. qb side is common and bp = qa = √2 (given). So if we can prove the included angle ∠qbp = ∠aqb, then the triangles can be proved to be congruent and thus pq = ab = 2 given.
 

1. What is a geometry proof?

A geometry proof is a logical argument that uses previously established theorems and postulates to show the validity of a statement or theorem in geometry. It is a step-by-step process that demonstrates the truth of a given statement.

2. What are some tips for writing a geometry proof?

Some tips for writing a geometry proof include clearly stating the given information, identifying what needs to be proved, and using correct notation and labeling. It is also important to use logical reasoning and provide justification for each step in the proof.

3. What are some common theorems used in geometry proofs?

Some common theorems used in geometry proofs include the Pythagorean Theorem, the Angle Sum Theorem, and the Triangle Congruence Theorems (SSS, SAS, ASA, AAS, HL). Other important theorems include the Parallel Lines Theorem and the Similarity Theorems (AA, SAS, SSS).

4. How can I check the validity of a geometry proof?

To check the validity of a geometry proof, you can follow along with each step and make sure that it follows logically from the given information and previously established theorems. You can also try to come up with a counterexample to disprove the statement being proved.

5. What are some common mistakes to avoid in geometry proofs?

Some common mistakes to avoid in geometry proofs include assuming what needs to be proved, using incorrect notation or labeling, and making invalid logical jumps. It is also important to be thorough and provide justification for each step in the proof.

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