Mother’s umbrella

A circle containing two internally tangent circles. Prove that the red triangle is isosceles.

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This solution makes use of the lemma below, attributed to Archimedes, showing that F is the midpoint of the arc formed by the tangent chord.

Archimedes lemma


Please find here the video solution: Video: Mother’s umbrella.

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2 replies on “Mother’s umbrella”

Dilation of the green circle from the point L to the red circle shows that the horizontal blue tangents are parallel. Then the three angles marked alpha are equal, and the triangles CEM ~LEC are similar. Thus EL*EM = EC^2, and the power of E with respect to any “green-like” circle is the same. This means the radical axis for any pair of intersecting “green-like” circles must go through the point E.

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