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An equilateral triangle inscribed in a rectangle

+ + + +
+ + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + +
+ +Slide the “@” up and down to change the geometry.
+Press “r” to reset the diagram to its initial state.
+Proposition: The blue area equals the sum of the two pink areas. + +
+ +

Problem statement

+ +

+The diagram above shows an equilateral triangle inscribed in a rectangle +in such a way that the two have a vertex in common. This subdivides the +rectangle into four disjoint triangles. +The original equilateral triangle is shown in white +in the diagram; the other three are shown in color. + +

+Proposition + +The area of the blue triangle equals the sum +of the areas of the two pink triangles. + + +

+The trigonometric proof is quite straightforward. I don't +know of a classical proof a la Euclid. +(Well, actually I haven't tried much.) +If you can think of a neat non-trigonometric proof, let me know. I will +put it here with due credit. + +

+This problem appeared as a conjecture +in an article +in the geometry.puzzles newsgroup on March 15, 1997. + +

+Note added January 8, 2017: +Here is a +clever solution +that Peter Renz sent me a in December 2016. Thanks, Peter! + +

+

+This applet was created by +Rouben Rostamian +using +David Joyce's +Geometry +Applet +on July 2, 2010. + +

+ + + + + +
Go to Geometry Problems and Puzzles + +Valid HTML + +Valid CSS +
+ + + + diff --git a/public/style.css b/public/style.css new file mode 100644 index 0000000..9395a45 --- /dev/null +++ b/public/style.css @@ -0,0 +1,56 @@ +body { + /* margin:35px auto; */ + /* width: 780px; */ + margin-top: 35px; + margin-left: 10%; + margin-right: 10%; + margin-bottom: 30px; + line-height: 140%; + font-size: 110%; + /* font-family: Verdana, sans-serif; */ + /* background: #ff8247; */ +} + +dt { + font-weight: bold; +} + +dd { + margin-bottom: 2ex; +} + +li { + margin-bottom: 1.0ex; +} + +.smallersize { + font-size: smaller; +} + +table.centered { + margin: auto; +} + +p.centered { + text-align: center; +} + +div.centered { + text-align: center; +} + +span.name { + font-variant: small-caps; +} + +img.noborder { + border-width: 0px; +} + +div.highlight { background: yellow } + +span.highlight { background: yellow; } + +span.strike { text-decoration: line-through; } +span.uline { text-decoration: underline; } +