** Write-Up #4**:

__Part A: Investigations__

The **centroid(G)** of a triangle **is
the common intersection of the three medians**. A median of
a triangle is the segment from a vertex to the midpoint of the
opposite side.

The **orthocenter (H)** of a triangle **is
the common intersection of the three lines containing the altitudes**.
An altitude is a perpendicular segment from a vertex to the line
of the opposite side. (Note: H may not always be in the triangle.)

The **circumcenter (C)** of a triangle **is
the point in the plane equidistant from the three vertices of
the triangle**. Since a point equidistant from two points lies
on the perpendicular bisector of the segment determined by the
two points, **C is on the perpendicular bisector of each side
of the triangle and is the center of the **** circumcircle** (the circumscribed
circle) of the triangle. (Note: C many be outside of the triangle.)

The **incenter **(I) of a triangle **is
the point on the interior of the triangle that is equidistant
from the three sides. **Since a point interior to an angle that**
**is equidistant from the two sides of the angle lies on the
angle bisector, then the **Incenter (I) lies on the angle bisector
of each angle of the triangle**.** **The **incircle**
(inscribed circle) must use the shortest distance to the sides;
hence the **radius of the incircle is along the orthogonal line**.

The **medial triangle** is a triangle that
connects the three midpoints of the sides. It is similar to the
original triangle and is 1/4 of its area.

The **orthic triangle** is an **acute**
triangle that connects the feet of the altitudes.

The **Nine-Point Circle** is a circle that
passes through the three mid-points of the sides, the three feet
of the latitudes, and the three mid-points of the segments from
the respective vertices to orthocenter, and has center at N, which
corresponds to the Circumcenter, C.

__Part B: Investigation of the Orthocenter
of a Triangle__

The orthocenter of a triangle is the common intersection of the three lines containing the altitudes. An altitude is a perpendicular segment from a vertex to the line of the opposite side and may be external to the triangle.

Given a triangle, construct one altitude from the top vertex of a triangle as shown.

In a similar manner construct the other three orthogonal projections from the other vertices:

Now if we hide the construction lines (ie, the altitude lines), then we can observe the orthocenter H, and the effects of different situations with an obtuse triangle. From the above example, the orthocenter is inside the triangle since the projections are inside the triangle.

If the triangle is obtuse (ie if the left vertex is an obtuse angle - an angle between 90 and 180 degrees), then the orthocenter is observed to be outside of the triangle because the (orthogonal) projection falls outside the triangle as shown below.

If we observe the loci of the orthocenter while the upper vertex traces a line from left to right, then H traces an arc as follows:

Hence, the orthocenter falls inside the triangle for acute angles and outside the triangle for obtuse angles.

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