## Congruent triangles

Let’s prove one direction: (1). Consider shown below:

We showed previously that in any triangle with orthocenter . Thus . By one of the equivalent statements here, we conclude that equation (1) holds.

Similar to example 1 above. If is the reflection of the circumcenter over side , then we always have in any triangle. This would then lead to , which by the equivalent statements here implies equation (1).

This follows from the two preceding examples, together with the fact that the segment has length . (Recall that is congruent to the parent in such a way that , , and .)

## Cyclic trapezium

Note that under the assumption of equation (1), we have that the reflection lies on side . Let be the circumradius of the parent triangle . Then and also . Since radius is parallel to side (see here), we get an isosceles trapezium .

Similar to example 4 above.

## Takeaway

In , let be the side-lengths, the circumcenter, the reflection of over side , the reflection of over side , and the reflection of over side . Then the following statements are *equivalent*:

As usual, we have more of these in the exercises.

## Task

- (Mid 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-six*statements are*equivalent*:- is congruent to
- is isosceles with
- is isosceles with
- is right angled at
- is the circumcenter of
- is right-angled at
- is right-angled at
- quadrilateral is a rectangle
- the points are concyclic with as diameter
- the reflection of over lies internally on
- the reflection of over lies externally on
- radius is parallel to side
- is the reflection of over side
- the nine-point center lies on
- the orthic triangle is isosceles with
- the geometric mean theorem holds
- the bisector of has length , where
- the orthocenter is a reflection of vertex over side
- segment is tangent to the circumcircle at point
- median has the same length as the segment
- the bisector of is tangent to the nine-point circle at
- is a convex
*kite*with diagonals and - altitude is tangent to the nine-point circle at
- 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 .