Teaser 1885: Sacred sign
From The Sunday Times, 1st November 1998 [link]
The “Sacred Sign of Solomon” consists of a pentagon whose vertices lie on a circle, roughly as shown:
Starting at one of the angles and going round the circle the number of degrees in the five angles of the large pentagon form an arithmetic progression; i.e., the increase from the first to the second equals the increase from the second to the third, etc.
As you can see, the diagonals form a small inner pentagon and in the case of the sacred sign one of its angles is a right angle.
What are the five angles of the larger pentagon?
This puzzle was included in the book Brainteasers (2002, edited by Victor Bryant). The puzzle text above is taken from the book.
[teaser1885]
S2T2 
Jim Randell 8:32 am on 1 September 2020 Permalink |
If the five sides of the pentagon are A, B, C, D, E, then the angles subtended by each side at the circumference of the circle are all equal, say, a, b, c, d, e.
(The angles in the small pentagon, such as ace denote the sum of the three symbols, i.e. (a + c + e) etc).
We see that: a + b + c + d + e = 180° (from many triangles, cyclic quadrilaterals, and the angles subtended at the centre of the circle).
The internal angles of the pentagon are an arithmetic progression, say: (x − 2y, x − y, x, x + y, x + 2y).
So, let’s say:
And these angles sum to 540°.
And if: x = b + c + d = 108°, then: a + e = 72°
So starting with: a + d + e = x + 2y, we get:
Similarly we express each of the angles a, b, c, d, e in terms of y:
The internal angles of the small pentagon are:
And one of these is 90°.
Our candidates are (b + c + e) and (b + d + e).
And the angles of the large pentagon are then:
For the other candidate:
This is not a viable solution, so:
Solution: The angles of the larger pentagon are: 96°, 102°, 108°, 114°, 120°.
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Jim Randell 11:54 am on 1 September 2020 Permalink |
Here’s a drawing of the symbol, with the two perpendicular lines indicated:
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