To find the orthocenter of each triangle in this exercise, you need to follow these steps:

1. Draw the Altitudes: For each triangle, an altitude is a line segment from one vertex perpendicular to the opposite side. Draw the three altitudes in each triangle; these will intersect at the orthocenter.

2. Find the Intersection Point: The orthocenter, labeled $P$, is the point where the three altitudes intersect.

Now, let's go through each triangle type:

• Triangle 19: This appears to be an acute triangle (all angles are less than 90°). In acute triangles, the orthocenter lies inside the triangle. Draw perpendicular lines from each vertex to the opposite side, and find the intersection point within the triangle.

• Triangle 20: This triangle is also acute. Similar to the previous triangle, draw all three altitudes, and the orthocenter will be located inside the triangle.

• Triangle 21: This is a right triangle (one angle is 90°). For right triangles, the orthocenter is at the vertex of the right angle. Therefore, point $C$ is the orthocenter.

• Triangle 22: This is an obtuse triangle (one angle is greater than 90°). For obtuse triangles, the orthocenter lies outside the triangle. Extend the altitudes from each vertex, and you will find the orthocenter outside the triangle.

Would you like further details or help with the calculations for each altitude?

Here are some related questions for further exploration:

1. What properties distinguish the orthocenter from the centroid or circumcenter in different types of triangles?
2. How does the location of the orthocenter vary in obtuse, acute, and right triangles?
3. Can a triangle's orthocenter, centroid, and circumcenter ever coincide? If so, under what conditions?
4. What role does the orthocenter play in the broader study of triangle centers?
5. How do you construct altitudes accurately using a compass and straightedge?

Tip: In a right triangle, the orthocenter is always located at the vertex of the right angle.