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# Nine-point center equals a vertex II

Among all triangles, the only triangle whose nine-point center coincides with one of its vertices is the -triangle.

As such, it has unique properties, some of which will be shown in today’s post.

Additionally, the -triangle is also unique among all isosceles triangles, and we’ll see that later.

PROVE that in the -triangle, the two equal sides are equal to the circumradius.

Let be the circumradius and let be the side-lengths of a given triangle . By the extended law of sines we have:

In the event that , we get .

If one vertex of a triangle is equidistant from the midpoints of its sides, PROVE that the triangle is the -triangle.

Suppose that vertex of triangle is equidistant from the midpoints of sides . In particular, this means that the median from vertex is equal to half the side-lengths of and .

By the cosine formula:

Consequently, .

PROVE that in the -triangle, one vertex is equidistant from the midpoints of the three sides.

Note that the side-lengths are in the ratio , and so . Consider the length of the median from vertex :

And so vertex is equidistant from the midpoints of all three sides.

If one vertex of a triangle is equidistant from the feet of its three altitudes, PROVE that the triangle is the -triangle.

Let the feet of the altitudes be , and let the altitudes themselves be . Suppose that vertex is equidistant from and , that is . Then:

Similarly, let :

And so . Since is equivalent to , we obtain . In turn, . The fact that then gives .

PROVE that in the -triangle, one vertex is equidistant from the feet of all three altitudes.

Easy converse to the preceding example.

## Takeaway

In any triangle , let be the side-lengths and the circumradius. Then the following statements are equivalent:

1. is equidistant from the midpoints of the sides
2. is equidistant from the feet of the three altitudes
3. is equidistant from the three Euler points.

Such must be the nine-point center.

• (Late seventies) In a non-right triangle , let be the side-lengths, the altitudes, the feet of the altitudes from the respective vertices, the midpoints of sides , the Euler points, the circumradius, the circumcenter, the nine-point center, the orthocenter, the reflection of over side , the reflection of over side , and the reflection of over side . PROVE that the following seventy-eight statements are equivalent:
1. or
2. is the reflection of over side
3. is the reflection of over side
4. is congruent to
5. is congruent to
6. is isosceles with
7. is isosceles with
8. is right angled at
9. is the circumcenter of
10. is right-angled at
11. is right-angled at
13. the points are concyclic with as diameter
14. the reflection of over lies internally on
15. the reflection of over lies externally on
16. radius is parallel to side
17. is the reflection of over side
18. segment is perpendicular to side
19. the nine-point center lies on
20. the orthic triangle is isosceles with
21. the geometric mean theorem holds
22. the bisector of has length , where
23. the orthocenter is a reflection of vertex over side
24. segment is tangent to the circumcircle at point
25. median has the same length as the segment
26. the bisector of is tangent to the nine-point circle at
27. is a convex kite with diagonals and
28. altitude is tangent to the nine-point circle at
29. chord is a diameter of the nine-point circle
30. segment is tangent to the nine-point circle at .
(Almost losing count with more still lined up.)