The diagram shows a regular pentagram. Circles of the same color are congruent. Wherever things look tangent, they are tangent.
Are the red and green circles congruent?
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$\begingroup$ How do you come up with all of these (interesting puzzles)? $\endgroup$Plop– Plop2025-02-26 12:35:55 +00:00Commented Feb 26 at 12:35
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1$\begingroup$ @Plop I play with circles and other shapes on my computer. If I find some interesting property, for example a "perfect fit", then I try to find an intuitive explanation. $\endgroup$Dan– Dan2025-02-26 12:57:43 +00:00Commented Feb 26 at 12:57
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1 Answer
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Answer:
Yes
Explanation:
Draw the two blue lines tangent to the top green circles and the bottom green circle. The yellow lines are two of the lines in the star that are tangent to the top red circle and the top green circles. The blue lines are parallel to the yellow lines, and by symmetry, they form a rhombus. The purple line (the horizontal diagonal of the rhombus) passes through the centres of the top green circles. Therefore, the two red circles and the central and bottom green circles are related by reflection about the purple line.
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6$\begingroup$ Why would the blue lines be parallel to the yellow ones? Seems to me that's entirely dependent on whether all the circles are the same size or not, which is what we're supposed to decide. $\endgroup$Bass– Bass2025-02-16 11:26:52 +00:00Commented Feb 16 at 11:26
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6$\begingroup$ @Bass I agree the argument is lacking, but it is easily fixed. The green circles are arranged in in 5-fold symmetry, so the blue lines are also sides of a smaller pentagram tangent to the green circles. $\endgroup$Jaap Scherphuis– Jaap Scherphuis2025-02-16 12:21:12 +00:00Commented Feb 16 at 12:21
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2$\begingroup$ @Bass. What Jaap said was what I had in mind. There’s so much symmetry in the problem that I didn’t think it was necessary to explain that step. $\endgroup$Pranay– Pranay2025-02-16 15:49:21 +00:00Commented Feb 16 at 15:49
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2$\begingroup$ Even with Jaap's comment I struggled at first to see how that proves anything. To future readers not getting it either I'd like to state the obvious: Because you could make another regular pentagram with the blue lines it must mean that the angle where the blue lines cross are identical to the angle where the yellow lines cross, proving that the blue lines and yellow lines are parallel to each other. (no matter the size of a regular pentagram, the angle at the tips is always the same) $\endgroup$Ivo– Ivo2025-02-17 08:41:42 +00:00Commented Feb 17 at 8:41