Proving two circles are orthogonal

In summary, the problem involves showing that the segment from the center of the smaller circle to point F forms a right angle with line segment CF. This can also be shown by demonstrating that line segment CH forms a right angle with the line segment from the center of the smaller circle to H. However, neither of these methods seem feasible since the center of the smaller circle is not included in the given image. The power of point E is EF*EA which is equal to EH*EB, and it is also noted that angles AFB and AHB are both right angles. The secant-secant angle theorem is not applicable in this case due to unknown intercepted arcs and the lack of inverses. Perhaps Pythagoras' theorem could be
  • #1
jdinatale
155
0

Homework Statement


prooffff.png

Homework Equations


The Attempt at a Solution



Here's an image of what I need to show.

pf2.png


I know I need to show that the segment from the center of the smaller circle to F forms a right angle with line segment CF. Alternatively I could show that line segment CH forms a right angle with the line segment from the center of the smaller circle to H. Both of those methods seem unlikely, since the center of the smaller circle isn't even included in the original image in my textbook.

I noticed the the Power of point E is EF*EA which equals EH*EB. I also noticed that [itex]\angle AFB = \angle AHB = 90^\circ[/itex]

I also have the secant-secant angle theorem, but that doesn't seem to help, because we don't know the measure of the intercepted arcs, plus that doesn't seem really useful anyways.

I also don't have access to inverses, so I can't use those.
 
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  • #2
Can you use Pythagoras' theorem?
 

Related to Proving two circles are orthogonal

1. How do you prove two circles are orthogonal?

To prove two circles are orthogonal, you need to show that the tangents to the circles at the point of intersection are perpendicular. This can be done by using the property that the tangent to a circle is perpendicular to the radius at the point of contact.

2. What is the definition of orthogonal circles?

Orthogonal circles are two circles that intersect at right angles. In other words, the tangents to the circles at the point of intersection are perpendicular.

3. Can two circles be orthogonal if they do not intersect?

No, two circles must intersect in order to be considered orthogonal. If two circles do not intersect, they are said to be disjoint and cannot be orthogonal.

4. Are all circles with equal radii orthogonal?

No, not all circles with equal radii are orthogonal. Two circles with equal radii can only be orthogonal if they intersect at right angles. Otherwise, they are just two equal circles that do not have an orthogonal relationship.

5. Can you prove two circles are orthogonal using their equations?

Yes, you can prove two circles are orthogonal by setting their equations equal to each other and solving for the points of intersection. Then, you can use the slope formula to show that the tangents at the point of intersection are perpendicular.

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