This is a paragraph.

# Seventy-five equivalent statements

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 . Then the following seventy-five 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 .

It will take quite some time to establish the above equivalence. In this post we select just five.

PROVE that implies .

This follows from the Pythagorean identity .

PROVE that implies , where is the radius of the circumcircle of triangle .

By the extended law of sines, we have that . And so:

The converse also holds.

PROVE that implies in a non-right triangle .

We use the identity and the extended sine law :

Since is non-right, take .

PROVE that implies .

PROVE that implies .

From example 3 we had . By the converse of example 1 . Thus .

## Takeaway

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

• (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 .
(Target reached! And surpassed!)
• (Extra feature) If satisfies equation (??), PROVE that its nine-point center divides in the ratio .