{"id":3158,"date":"2023-04-29T11:20:16","date_gmt":"2023-04-29T15:20:16","guid":{"rendered":"https:\/\/josmfs.net\/?p=3158"},"modified":"2025-07-22T15:49:21","modified_gmt":"2025-07-22T19:49:21","slug":"curious-sunbeam-problem","status":"publish","type":"post","link":"https:\/\/josmfs.net\/wordpress\/2023\/04\/29\/curious-sunbeam-problem\/","title":{"rendered":"Curious Sunbeam Problem"},"content":{"rendered":"

\"\"This is a Catriona Agg problem presented by itself, since it turned out to be the most challenging one I ever tried.\u00a0 Usually I can solve her problems in a few minutes or maybe hours, or sometimes days if they are especially challenging.\u00a0 But this problem has taken me weeks and I had to rely on a non-geometric argument.\u00a0 The problem is full of fascinating and unexpected relationships, but I couldn\u2019t find a way to use them to prove the answer.<\/p>\n

See the Curious Sunbeam Problem<\/a><\/p>\n

(Update 5\/5\/2023, 7\/22\/2025)\u00a0 Alternative Solutions<\/strong><\/p>\n

(Update 5\/1\/2023) Oscar Rojas Solution<\/strong><\/p>\n

\"\"<\/p>\n

Several people on Catriona Agg\u2019s twitter site mentioned cyclic quadrilaterals, but that is not such a familiar concept to me, other than being a quadrilateral whose vertices all lie on a circle.\u00a0 I went to Wikipedia to look at properties, of which there are many, and tried to find something that might be relevant.\u00a0 But I confess I failed, though I did not pursue it too extensively.\u00a0 My first problem is that it was not obvious to me that the four points B, C, E, and F all lie on a circle.\u00a0 Sure, any three could, but what about the fourth?\u00a0 What property explicitly was used to prove they were cyclic?\u00a0<\/p>\n

Another question I had was regarding the \u201coutside angle theorem\u201d.\u00a0 I admit I had not heard of that, or if I did, I did not remember it after more than 60 years.\u00a0 I did a Google search and found many references to the \u201cexternal angle theorem\u201d, which is obvious and common, but nothing about an \u201coutside angle theorem\u201d.\u00a0 So it would have been helpful to state it and perhaps give a reference.\u00a0 But of course I had found the angle was 45\u00b0 by a different method.\u00a0<\/p>\n

To have to resort to less elementary geometric properties makes the problem a bit harder, which Catriona admitted, though her approach was even different (and not immediately obvious to me).\u00a0 Clearly, I am an amateur when it comes to the vast repertoire of plane geometry, which might be expected for anyone regarding a subject that was developed over 2000 years.\u00a0 I just like how many geometric puzzles one can solve with the simplest ideas.<\/p>\n

(Update 7\/22\/2025) Ibo Beyazit Solution <\/strong><\/p>\n

Ibo Beyazit has provided another solution to the problem that doesn\u2019t involve cyclic quadrilaterals.\u00a0<\/p>\n

See Ibo Beyazit Solution<\/a><\/p>\n

A possible more direct line of argument in his solution might be Step 3 implies Step 7 (sides of large right triangle are equal).\u00a0 Then Step 6 implies the perpendicular lines cut the sides of the large right triangle in half, which implies Step 9 (the small right triangle above the square is isosceles), and thus the unknown angle is 45 degrees.<\/p>\n\n\n

<\/p>\n","protected":false},"excerpt":{"rendered":"

This is a Catriona Agg problem presented by itself, since it turned out to be the most challenging one I ever tried.\u00a0 Usually I can solve her problems in a few minutes or maybe hours, or sometimes days if they are especially challenging.\u00a0 But this problem has taken me weeks and I had to rely […]<\/p>\n","protected":false},"author":1,"featured_media":0,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"footnotes":""},"categories":[3],"tags":[42,13],"class_list":["post-3158","post","type-post","status-publish","format-standard","hentry","category-puzzles-and-problems","tag-catriona-agg","tag-plane-geometry"],"_links":{"self":[{"href":"https:\/\/josmfs.net\/wordpress\/wp-json\/wp\/v2\/posts\/3158","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/josmfs.net\/wordpress\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/josmfs.net\/wordpress\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/josmfs.net\/wordpress\/wp-json\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/josmfs.net\/wordpress\/wp-json\/wp\/v2\/comments?post=3158"}],"version-history":[{"count":6,"href":"https:\/\/josmfs.net\/wordpress\/wp-json\/wp\/v2\/posts\/3158\/revisions"}],"predecessor-version":[{"id":4482,"href":"https:\/\/josmfs.net\/wordpress\/wp-json\/wp\/v2\/posts\/3158\/revisions\/4482"}],"wp:attachment":[{"href":"https:\/\/josmfs.net\/wordpress\/wp-json\/wp\/v2\/media?parent=3158"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/josmfs.net\/wordpress\/wp-json\/wp\/v2\/categories?post=3158"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/josmfs.net\/wordpress\/wp-json\/wp\/v2\/tags?post=3158"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}