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Tangent Ellipses (Posted on 2018-11-08) Difficulty: 4 of 5
There is a circle of a radius √(8/3).

There are 3 ellipses inside the circle which are tangent to the circle at points A, B, C and to each other at points D, E, F. Their major axes are parallel to lines tangent to the circle at corresponding tangency points. The lengths of major axes of the ellipses are 2√(3/2), 2√(8/7) and 2.

What is the ratio of the area of triangle ABC to the area of triangle DEF?

No Solution Yet Submitted by Danish Ahmed Khan    
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Question A question | Comment 1 of 2
Dear DAK,
I am having trouble making the picture. I do not see how an ellipse within a circle can, all at the same time: be tangent to the circle, have a major axis parallel to that tangent, and still be completely within the circle. It seems to me that such an ellipse must intersect the circle and be partially outside the circle. 

To do what is required, it must have its _minor_ axis parallel to the tangent line, or else be a circle, and in the latter case, it would have no unique major axis. 

So, what am I missing?

Thanks very much,
Steve Lord

P.S., Here is my picture:

Edited on November 9, 2018, 2:44 am
  Posted by Steven Lord on 2018-11-09 02:42:26

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