This is a paragraph.

# An example of harmonic division

Due to a little -related issue, we chose to change the format of today’s post. Please click here for the pdf.

## Takeaway

Let and be the feet of the symmedian and the altitude (both) from vertex in triangle . Then the following statements are equivalent:

1. and divide harmonically
2. the identity holds.

• (Growing membership) 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 in that order, the Euler points, the circumradius, the circumcenter, the nine-point center, the orthocenter, the reflection of over side , the reflection of over side , the reflection of over side , the symmedian point, the foot of the symmedian from vertex , and the radius of the polar circle. PROVE that the following eighty-four statements are equivalent:
1. or
2. and divide harmonically
3. is the reflection of over side
4. is the reflection of over side
5. is congruent to
6. is congruent to
7. is isosceles with
8. is isosceles with
9. is right angled at
10. is the circumcenter of
11. is right-angled at
12. is right-angled at
14. the points are concyclic with as diameter
15. the reflection of over lies internally on
16. the reflection of over lies externally on
17. radius is parallel to side
18. is the reflection of over side
19. segment is perpendicular to side
20. the nine-point center lies on
21. the orthic triangle is isosceles with
22. the geometric mean theorem holds
23. the bisector of has length , where
24. the orthocenter is a reflection of vertex over side
25. segment is tangent to the circumcircle at point
26. median has the same length as the segment
27. the bisector of is tangent to the nine-point circle at
28. is a convex kite with diagonals and
29. altitude is tangent to the nine-point circle at
30. chord is a diameter of the nine-point circle
31. segment is tangent to the nine-point circle at .
(Once we hit , we’ll stop this particular exercise.)