An altitude in a triangle is a perpendicular
segment from a vertex to the side opposite that vertex. Below,
the three altitudes of **D ABC **are extended. Also,
the intersection points (feet) of the altitudes and the sides
of the triangle are labeled **O, R, **and **T**. The segments
connecting these feet of the altitudes form the ** orthic triangle, D
ORT. **Point

Recall that the ** circumcircle** of

You probably have a suspicion that **D XYZ **and **D ORT **have some relationships.
For example, they appear to be similar.

Here are a few observations that can be made. Click on the picture to investigate and verify these conjectures in GSP:

1. The orthic triangle

(D ORT)and circum-altitude triangle(D XYZ)are similar.

PROOF2. The area of the orthic triangle

(D ORT)is one-fourth the area of the circum-altitude triangle(D XYZ).

PROOF3. The perimeter of the orthic triangle

(D ORT)is one-half the perimeter of the circum-altitude triangle(D XYZ).

PROOF4. The orthic triangle

(D ORT)and circum-altitude triangle(D XYZ)have the same incenter.

PROOF5. The incenter of the orthic

(D ORT)and circum-altitude(D XYZ)triangles is the orthocenter of the original triangle.

PROOF