Problem 264

Triangle Centres

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)

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.

三角形的心

考虑满足如下条件的三角形：

各顶点都在格点上。
外心在原点O处。
垂心位于点H(5, 0)。

在边长≤ 50的所有三角形中，满足上述条件的三角形有九个。
按照它们的周长升序排列，分别是：

它们的周长之和四舍五入到四位小数是291.0089。

在边长≤ 10⁵的所有三角形中，找出所有满足上述条件的三角形。
将它们的周长之和四舍五入到四位小数作为你的答案。