Similarly, the two points in each of the other sides of the triangle ABC may be shewn to be on the circumference of the circle.
The circle which is thus shewn to pass through these nine points may be called the Nine points circle: it has some curious properties, of which we will now give two.
The radius of the Nine points circle is half of the radius of the circle described round the original triangle.
For the triangle DEF has its sides respectively halves of the sides of the triangle ABC, so that the triangles are similar. Hence the radius of the circle described round DEF is half of the radius of the circle described round ABC.
If S be the centre of the circle described round the triangle ABC, the centre of the Nine points circle is the middle point ofSO.
For HS is at right angles to BC, and therefore parallel to GO. Hence the straight line which bisects HG at right angles must bisect SO. And H and G are on the circumference of the Nine points circle, so that the straight line which bisects HG at right angles must pass through the centre of the Nine points circle. Similarly, from the other sides of the triangle ABC two other straight lines can be obtained, which pass through the centre of the Nine points circle and also bisect SO. Hence the centre of the Nine points circle must coincide with the middle point of SO.
We may state that the Nine points circle of any triangle touches the inscribed circle and the escribed circles of the triangle: a demonstration of this theorem will be found in the Plane Trigonometry, Chapter xxiv. For the history of the theorem see the Nouvelles Annales de Mathematiques for 1863, page 562.
31. If two straight lines bisecting two angles of a triangle and terminated at the opposite sides be equal, the bisected angles shall be equal.
Let ABC be a triangle; let the straight line BD bisect the angle at B, and be terminated at the side AC; and let the straight line CE bisect the angle at C, and be terminated at the side AB; and let the straight line BD be equal to the straight line CE: then the angle at B shall be equal to the angle at C
