# Three Right Triangles

There are three special right triangles, all of their edge lengths are positive integers.

One of the triangles' longer leg, the other triangle's shorter leg and hypotenuse of the last triangle are all equal to each other.

What is the least value of the shortest hypotenuse among these triangles?

15

it can be found in these triangles:

9 12 15
8 15 17
15 20 25

note that:

two of these are scalings of the smallest triangle possible, the 3 4 5

Why smaller numbers do not work:

the smallest possible hypotenuses less than 15 are: 5, and 10, 13. 5 can only ever be in the hypotenuse, as it is so small. for 10 and 14, here are all the possible triangles with them:

10:
10 24 26
6 8 10

13:
5 12 13
13 84 89

as you can see:

none of these can be in the larger leg

Just a quick proof that numbers less than

15

don't work.

9 12 15
8 15 17
15 20 25 means 15 works as noted in micsthepick's answer

No more spoilers from here on - read at your own peril

The formula for primitive pythagorean triples is actually quite well-known. Now, that means that the largest number in a triple can be expressed as a multiple of a sum of squares, so that means the answer, if less than 15, is either 5,10 or 13.

## 5

$5^2+1^2=26$ is not a square

$5^2+2^2=29$ is not a square

$5^2+3^2=34$ is not a square

$5^2+4^2=41$ is not a square

## 10

$10^2+1^2=101$ is not a square

$10^2+2^2=104$ is not a square

$10^2+3^2=109$ is not a square

$10^2+4^2=116$ is not a square

$10^2+5^2=125$ is not a square

$10^2+6^2=136$ is not a square

$10^2+7^2=149$ is not a square

$10^2+8^2=164$ is not a square

$10^2+9^2=181$ is not a square

## 13

$13^2+1^2=170$ is not a square

$13^2+2^2=173$ is not a square

$13^2+3^2=178$ is not a square

$13^2+4^2=185$ is not a square

$13^2+5^2=194$ is not a square

$13^2+6^2=205$ is not a square

$13^2+7^2=218$ is not a square

$13^2+8^2=233$ is not a square

$13^2+9^2=250$ is not a square

$13^2+10^2=269$ is not a square

$13^2+11^2=290$ is not a square

$13^2+12^2=313$ is not a square