Henry Ernest Dudeney/Puzzles and Curious Problems/162 - Find the Triangle/Solution

by : $162$

 * Find the Triangle

Solution

 * The sides of the triangle are $13$, $14$, and $15$, making $14$ the base, the height $12$, and the area $84$.

Proof
As put it:


 * There is an infinite number of rational triangles composed of three consecutive integers, like $3$, $4$ and $5$, and $13$, $14$, and $15$,
 * but there is no other case in which the height will comply with our conditions.

Rational triangles whose sides are three consecutive integers with integer area are given by the recurrence relation:


 * $U_n = \begin {cases} 4 & : n = 1 \\ 14 & : n = 2 \\ 4 U_{n - 1} - U_{n - 2} & : n > 2 \end {cases}$

where the lengths of the sides of triangle $n$ are $U_n - 1$, $U_n$ and $U_n + 1$.

This can be presented as follows:

Let lengths of the sides of a triangle with integer area be $2 x - 1$, $2 x$ and $2 x + 1$.

Then $3 \paren {x^2 - 1}$ is a square number.

This is demonstrated in Approximations to Equilateral Triangles by Heronian Triangles.

Hence the rational triangles with integer area have the sides whose lengths are:


 * $\begin{array} {rrr}

3 & 4 & 5 \\ 13 & 14 & 15 \\ 51 & 52 & 53 \\ 193 & 194 & 195 \\ 723 & 724 & 725 \end {array}$

From Heronian Triangle whose Altitude and Sides are Consecutive Integers, the only one fitting the condition is the $13$, $14$, $15$ one.

It remains to exclude the possibility that we have note overlooked the possibility of a triangle whose sides are in the pattern $n$, $n + 1$, $n + 3$ with the height $n + 2$, for example.