{"app":{"locale":"en-GB"},"problem":{"content":"

Consider 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)

$\"p264_TriangleCentres.gif\"$

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

Find all such triangles with a perimeter ≤ 10⁵.
Enter as your answer the sum of their perimeters rounded to four decimal places.

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