## Triangle Centres

### Problem 264

Published on Saturday, 14th November 2009, 05:00 am; Solved by 409Consider all the triangles having:

- All their vertices on lattice points.
- Circumcentre at the origin O.
- Orthocentre at the point H(5, 0).

There are nine such triangles having a perimeter ≤ 50.

Listed and shown in ascending order of their perimeter, they are:

A(-4, 3), B(5, 0), C(4, -3) A(4, 3), B(5, 0), C(-4, -3) A(-3, 4), B(5, 0), C(3, -4) A(3, 4), B(5, 0), C(-3, -4) A(0, 5), B(5, 0), C(0, -5) A(1, 8), B(8, -1), C(-4, -7) A(8, 1), B(1, -8), C(-4, 7) A(2, 9), B(9, -2), C(-6, -7) A(9, 2), B(2, -9), C(-6, 7) |

The sum of their perimeters, rounded to four decimal places, is 291.0089.

Find all such triangles with a perimeter ≤ 10^{5}.

Enter as your answer the sum of their perimeters rounded to four decimal places.