The point at which the altitudes intersect in a triangle

Complete step by step solution:
We have been asked to find the point of concurrency of three altitudes of a triangle.
We know that the altitude of a triangle is a perpendicular drawn from a vertex to its opposite side. So, it is clear to us that since there are three vertices, there will be a total of three altitudes in a triangle.

Note: Remember the other terms related to a triangle like incenter, circumcenter and centroid and their definition, so that you can easily solve this kind of question. Now, we call the intersection point as incentre when all the angle bisectors of the triangle meet at that point. If we consider a circle circumscribing a triangle and if all the perpendicular bisectors of sides of the triangle meet at one point, then we call it the circumcentre. When all the medians of the three sides of a triangle meet at one point, we call that point a centroid. Also, remember that the point of concurrency means a point where all the lines, at least three intersect at a common point.

The centroid 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. So I first took midpoints of each side, then constructed segment from opposite vertex.The geometric centroid (center of mass) of the polygon vertices of a triangle is the point which is also the intersection of the triangle's three triangle medians. The point is therefore sometimes called the median point. The centroid is always in the interior of the triangle.

2. Orthocenter

The altitude of a triangle is a line which passes through a vertex of the triangle and is perpendicular to the opposite side. There are therefore three altitudes possible, one from each vertex. It turns out that all three altitudes always intersect at the same point - the so-called orthocenter of the triangle. The orthocenter is not always inside the triangle. If the triangle is obtuse, it will be outside. To make this happen the altitude lines have to be extended so they cross. Adjust the figure above and create a triangle where the orthocenter is outside the triangle. Follow each line and convince yourself that the three altitudes, when extended the right way, do in fact intersect at the orthocenter.

3. Circumcenter

The three perpendicular bisectors all come together in one point, called the circumcenter of triangle ABC. The circumcenter is equidistant from the three vertices, and so the common distance is the radius of a circle that passes through the vertices. It is called the circumcircle.

The orthocenter is the point of concurrency of the three altitudes of a triangle.  To understand what this means, we must first determine what an altitude is.  An altitude is a line that passes through a vertex of a triangle and that is perpendicular to the line that contains the opposite side of said vertex.  Below are two examples of altitudes, one in which the altitude is perpendicular to the opposite side, and one in which the altitude is perpendicular to the line that contains the opposite side.

Here we have an altitude that goes through vertex B and is perpendicular to side AC.  Below we will look at the second scenario, an altitude that goes through a vertex and that is perpendicular to the line that contains the opposite side, rather than the side itself.

In the image shown above, we see that there is no such line that passes through vertex B and is perpendicular to side AC.  In this instance, the altitude goes through vertex B and is perpendicular to the dotted line that contains side AC.  The altitude is pictured in red.  What type of triangle created the situation in which the altitude could not intersect the triangle�s side?

Now that we have a better grasp on what an altitude is, we are ready to begin discussing the orthocenter.  Every triangle has three altitudes, one that runs through each vertex.  When all three altitudes are drawn on the same triangle, they intersect at exactly one point, the orthocenter.  For a tool that allows you to test this, click HERE.

We will now examine the orthocenter of three different triangles:  acute, right, and obtuse.  If you explored the tool that creates the three altitudes, you may have already been able to make some conjectures about how the orthocenter changes.

First we will look at an acute triangle.  For any acute triangle, the three altitudes of the triangle intersect a vertex and opposite side of the triangle.  Therefore, for an acute triangle the orthocenter will always be on the interior of the triangle.  One such example is show below.

Next we will look at an obtuse triangle.  For any obtuse triangle two of the altitudes will intersect the line containing the opposite side, not the side itself.  This causes the orthocenter to lie outside the triangle.  One such example is shown below.

Lastly we will examine the right triangle.  Due to the fact that two of the legs of the right triangle form a right angle and pass through vertices of the triangle, the orthocenter of a right triangle will always lie on the vertex of the triangle�s right angle.  We see this in the image below.

Using what we�ve explored about the orthocenter of various triangles, try and determine the following.