Did you solve it? The smallest biggest triangle in the world

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Did you solve it? The smallest biggest triangle in the world

Did you solve it? The smallest biggest triangle in the world, Earlier today I asked you to construct a triangle whose existence seems to defy reason.

Show that there is a triangle, the sum of whose three heights is less than 1mm, that has an area greater than the surface of the Earth (510m km²).

Solution

Here’s one:

The triangle is isosceles (meaning two of the sides have the same length), has a height of 0.2mm and a base that is a few hundred light-years long.

I’ll take you through slowly how I got there. The area of a triangle is half the base times the height. Since there are three possible bases (and heights) there are thus three ways of describing the area of the same triangle. If triangle T has side lengths a, b, and c, and h_a is the height from side a to the opposite vertex, h_b is the height from side b to the opposite vertex and h_c is the height from side c to the opposite vertex, then the area of T can be expressed as either (ah_a)/2, (bh_b)/2 or (ch_c)/2, which all have the same value.

Let’s write this mathematically:

(ah_a)/2 = (bh_b)/2 = (ch_c)/2,

From this we can deduce that h_b = (a/b)h_a ,and likewise that h_c = (a/c)h_a.

Thus, for any triangle T, we can write the sum of all three heights in terms of the sum of one height:

h_a + h_b + h_c = h_a + (a/b)h_a + (a/c)h_a.

Now for the clever bit. Let’s imagine that T is an isosceles triangle, and that sides b = c. The fraction a/b is always going to be less than 2. We can see this by looking at the diagram below. When b (and c) is longer than a, as in the figure on the left, then a/b is less than 1. As b (and c) gets shorter and shorter, it will only get to half the size of a when b lies along a, and the triangle disappears. Thus the ratio a/b (and a/c) never reaches 2.

In other words, for an isosceles triangle T,

h_a + h_b + h_c < h_a + 2h_a + 2h_a = 5h_a

Translated into English, this means that for any isosceles triangle, the sum of its three heights is always less than five times the height measured from the ‘unequal’ side, irrespective of the lengths of the sides (since they are not mentioned in the equation).

As a consequence, we can make the sum of the heights of T as arbitrarily small as possible, because we can make the height from the ‘unequal’ side to the opposite vertex as arbitrarily small as we like. We can make the area of T as arbitrarily large as we like, since the area of T is 1/2 x base x height, and so all we need to do is chose a very large base to compensate for the small (but finite) height.

For example, if consider a triangle T below in which h_a = 0.2mm. Since 5h_a = 1mm, we have the situation mentioned in the question, which is that the sum of the three heights is less than 1mm.

Now we need to find a value for a, such that the area of T is larger than the area of the Earth (510,000,000 km²).

In other words, such that 1/2 x a x 0.2mm > 510,000,000km².

In fact, let’s say that 1/2 x a x 0.2mm = 511,000,000km², since this value works.

0.2mm = 0.0000002 km

Which gives us a = 5,110,000,000,000,000 km

I hope you enjoyed today’s puzzle. I’ll be back in two weeks.

Thanks again to Trần Phương, the Vietnamese maths guru who devised the puzzle.

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