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The envelope is a regular pentagon with two diagonals. The red coins are two vertically aligned and touching congruent circles, the top one passing through the apex of the pentagon, and the bottom passing through the intersection of the diagonals. The blue half-coin is a semicircle inscribed in the small triangle formed by the diagonals and the base of the pentagon.

Regular pentagon ABCDE, with base AB and two diagonals AC and BE which meet at F. The top red circle passes through D, and the bottom red circle passes through F. The blue semicircle is tangent to the diagonals with its base on AB.

If you slide the red coins vertically down until the bottom red coin touches the blue half-coin, will the top red coin be contained within the envelope (the pentagon)?

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    $\begingroup$ Still not done with pentagons and circles huh? 😄 $\endgroup$ Commented Nov 20, 2025 at 2:22
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    $\begingroup$ @justhalf. It’s a gift that keeps on giving ;) $\endgroup$ Commented Nov 20, 2025 at 2:25

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Pentagon vertices labeled A, B, C, D, E counterclockwise starting from bottom-left. Intersection of AC and BE labeled F.

The diameter of the top circle is the distance between D and EC, and the diameter of the bottom circle is the distance between B and AC. These distances are equal, so the circles are congruent.
added four lines: EC, line through D parallel to EC, lines through A and B parallel to BE and AC respectively.

If you slide the bottom circle up to touch the middle circle, it will pass through F. If you slide the top circle down by the same amount, it will be tangent to DE and DC the same way the bottom circle is tangent to FA and FB.

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  • $\begingroup$ That’s exactly how I proved it! $\endgroup$ Commented Nov 20, 2025 at 1:56
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    $\begingroup$ I added a figure to make the reveal spoilerequality of diameters manifest. Feel free to remove it if you don’t want it. $\endgroup$ Commented Nov 20, 2025 at 14:33
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    $\begingroup$ @Pranay hold on since when spoilers work in comments? is this the death of rot13? $\endgroup$ Commented Nov 20, 2025 at 15:35
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    $\begingroup$ @htmlcoderexe. Quite recently. Here’s the meta post on it: puzzling.meta.stackexchange.com/q/7741/95437 $\endgroup$ Commented Nov 20, 2025 at 15:35
  • $\begingroup$ @htmlcoderexe: you may still use rot13 if you so desire. The new spoiler feature is provided for convenience and beauty. And considering that the markdown is not advertised properly, and considering that the markup code is not similar enough with other markup codes, rot13 might come handier to some people. If one cannot decode rot13 on this site, then this site is probably not for them :) $\endgroup$ Commented Nov 21, 2025 at 9:23

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