Heron and his triangle

Geometry Level 4

The triangle with sides 6, 8, 10 satisfy the following properties:

All its side lengths are integers.
Both its area and its perimeter are equal integers.

Find another triangle with a minimal perimeter that satisfies these properties as well.

Submit your answer as the concatenation of these side lengths in increasing order.

For example, if you think that the side lengths of this other triangle is 10, 11, 12, then submit your answer as the 6-digit integer, 101112.

The answer is 51213.

1 solution

Kunal Gupta
Dec 4, 2016

These types of triangles are known as Equable Heronian Triangles . There are exactly five equable Heronian triangles: the ones with side lengths $(5,12,13)$ , $(6,8,10)$ , $(6,25,29)$ , $(7,15,20)$ , and $(9,10,17)$ .
The one with minimal perimeter is: $5,12,13$ making the answer to be : $51213$

Nice observation!

Michael Huang - 4 years, 6 months ago

How can we prove that there are only these 5 cases?

Calvin Lin Staff - 4 years, 6 months ago

Actually, we can see it from the following: $\text{Inradius}(r) = \dfrac{\Delta}{s}$ Here, $\Delta$ is the Area and $s$ is the semiperimeter.
If both area and semiperimeter are to be equal, then $r=2$ It turns out that for inradius of $2$ , and with integer sides, and integer area, there are only 5 cubic equations possible, ( these can be derived from using Heron's Formula, $r=2$ and using the fact that area is an integer ) giving the required sides.
The fact that $r=2$ in a sense limits the lengths of the triangles, ( otherwise one may be tempted to think that why only these sides, why not , say near $~ 1000$

Kunal Gupta - 4 years, 6 months ago

Great! Can you add these into the solution, and also state the equation that we derive from $r= 2$ ? Someone else could then help solve the diophantine equation when they see it.

Calvin Lin Staff - 4 years, 6 months ago

Heron and his triangle

The answer is 51213.

1 solution

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