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# Three special obtuse triangles I

We’ll be considering three obtuse triangles whose interior angles are:

• ,
• ,
• ,

What makes them special is the fact that their side-lengths satisfy the modified Pythagorean identity

(1)

with the accompanying equivalent descriptions. The first triangle has already appeared on August 14 and on October 14. The second triangle is making its debut here.

Suppose that the interior angles of are , . PROVE that the side-lengths satisfy equation (1).

Since , the conclusion follows from one of the equivalent statements here.

Suppose that the interior angles of are , . PROVE that satisfy .

In example 1 above we saw that the side-lengths satisfy equation (1). As such , by one of the equivalent statements here. So:

A direct computation of the side-lengths and can also be used to prove the above relationship, but that’s a longer procedure.

Suppose that the interior angles of are , . PROVE that the radius of the circumcircle of is the geometric mean of and ; that is, .

By example 2 above we had . Re-write as . Using one of the equivalent statements here, we know that .

One can also prove this by a direct computation of the circumradius and the fact that . Definitely a longer procedure.

Suppose that the interior angles of are , . PROVE that the side-lengths satisfy , and .

Let’s use the cosine formula and the fact that :

To show that , re-write as and then use the fact that just proved.

Find coordinates for the vertices of a triangle whose interior angles are , .

Take the tiny triangle where the vertices are located at , , and .

## Takeaway

Let be the side-lengths of , and let be the radius of its circumscribed circle. If equation (1) holds, then the following statements are equivalent:

1. .

Note the first one.

• (Late fifties) In a non-right triangle , let be the side-lengths, the altitudes, the feet of the altitudes from the respective vertices, the circumradius, the circumcenter, the nine-point center, the orthocenter, the midpoint of side , the reflection of over side , the reflection of over side , and the reflection of over side . PROVE that the following fifty-eight statements are equivalent:
1. or
2. is congruent to
3. is congruent to
4. is isosceles with
5. is isosceles with
6. is right angled at
7. is the circumcenter of
8. is right-angled at
9. is right-angled at
11. the points are concyclic with as diameter
12. the reflection of over lies internally on
13. the reflection of over lies externally on
14. radius is parallel to side
15. is the reflection of over side
16. the nine-point center lies on
17. the orthic triangle is isosceles with
18. the geometric mean theorem holds
19. the bisector of has length , where
20. the orthocenter is a reflection of vertex over side
21. segment is tangent to the circumcircle at point
22. median has the same length as the segment
23. the bisector of is tangent to the nine-point circle at
24. is a convex kite with diagonals and
25. altitude is tangent to the nine-point circle at
26. segment is tangent to the nine-point circle at .
( short of the target.)
• (Extra feature) If satisfies equation (??), PROVE that its nine-point center divides in the ratio .