Brain-Teaser 222: British triangles
From The Sunday Times, 25th July 1965 [link]
British Triangles, naturally, stand on horizontal bases with their points upwards. All their sides, never more than a sensible 60 inches, and their heights measure an exact number of inches. No B.T. is isosceles or right angled.
You can often put more than one B.T. on a common base. On a base of 28 inches 8 B.T.s are erected.
What are their heights?
This puzzle was included in the book Sunday Times Brain Teasers (1974, edited by Ronald Postill).
[teaser222]
Hugh Casement and 
S2T2 
Jim Randell 9:47 am on 27 November 2022 Permalink |
If the 3 sides of a triangle are a, b, c, then the area A is given by Heron’s formula:
And if the height (from side a) is h, then we also have:
Hence the height h can be determined from the sides a, b, c:
This Python program considers possible values for the 2nd and 3rd sides of the triangle, and then looks for values where the height is an integer.
It runs in 53ms. (Internal runtime is 1.8ms).
Run: [ @replit ]
from enigma import (irange, div, is_square, subsets, sq, printf) # check for a viable triangle def is_triangle(a, b, c): # check triangle inequalities if not (a + b > c and a + c > b and b + c > a): return # no triangle is isosceles if len({a, b, c}) < 3: return # no triangle is right angled sqs = (sq(a), sq(b), sq(c)) if sum(sqs) == 2 * max(sqs): return # looks OK return (a, b, c) # calculate integer height from side a def height(a, b, c): p = a + b + c x = div(p * (p - 2 * a) * (p - 2 * b) * (p - 2 * c), 4) if not x: return x = is_square(x) if not x: return h = div(x, a) return h # base a = 28 # consider possible integer length for the remaining sides, b, c for (b, c) in subsets(irange(1, 60), size=2, select="R"): if not is_triangle(a, b, c): continue # calculate the height of the triangle h = height(a, b, c) if not h: continue # output viable triangles printf("a={a} b={b} c={c} -> h={h}")Solution: The heights of the triangles are 9″ (2 triangles), 15″ (4 triangles), 24″ (2 triangles).
The four triangles found by the program are shown below. And they can be mirrored to produce the remaining four triangles.
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Hugh Casement 7:41 am on 28 November 2022 Permalink |
Alternatively we can think of it as two right-angled triangles joined together.
If their bases are d and e, then d² = b² – h² and e² = c² – h².
d + e, or |d – e| in the case of the obtuse-angled triangles, must equal a = 28.
b must not equal c, for then the overall triangle would be isosceles
(that is the case when h = 48, b = c = 50, and d = e = 14).
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Hugh Casement 9:09 am on 28 November 2022 Permalink |
Joined together along the line of their common height h, of course.
A base a = 21 also allows eight resultant triangles, including reflections.
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