Five points ABCDE are equally spaced around the dotted circle. A smaller red circle is tangent to (ABCDE) at a point F, between B and C. Blue and green tangents are drawn from ABCDE to the red circle, and are coloured blue and green as shown. Show that the length sum of the blue tangents equals the length sum of the green tangents.

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## Solution

The proof uses the Seven chords, but then for five chords. It also uses the below lemma that the tangent lengths are proportional to the large circle chords to F with a constant factor √(1-k).