GeoGebra
Name: ___________________________
5.2 Bisectors of Triangles and 5.3 Medians and Altitudes of Triangles
Part 1: Circumcenters
1. The circumcenter is the point of concurrency of the ____________________ of a triangle.
2. Draw any triangle on GeoGebra and construct its circumcenter. Be sure to add point P
to the center. To do this, use the ‘intersect two objects’ tool and make sure two of
the lines are highlighted before you click to place the point. If GeoGebra does not
choose the correct letter for you, right click on the letter and select ‘Rename.’ Enter
the correct letter.
3. Without moving the triangle, use the ‘Distance or Length’ tool to find the distance between the two
points.
AP = _______
BP = _______
CP = _______
4. Pick any vertex of your triangle and drag to change the shape of your triangle. Observe how the
circumcenter moves.
a. Where is the circumcenter located when the triangle is acute? (Inside the triangle, outside the
triangle, etc…)
___________________________________
b. … when the triangle is obtuse?
____________________________________
c. … when the triangle is right?
____________________________________
5. Using the numbers you discovered in question 3 to complete the theorem: The circumcenter of a
triangle is ____________ from the _____________ of a triangle.
6. IMPORTANT! Using the same triangle and circumcenter, locate the ‘circle with the center
through point’ tool. Use the circumcenter as the center of the circle. Pick any of the triangle vertices
as the second point. Why do you think the circumcenter is called the circumcenter?
_________________________________________________________________
Part 2: Incenters
1. The incenter is the point of concurrency of the ________________________ of a triangle.
2. Erase your drawing from Part 1 and start by drawing a new triangle. Construct its incenter. Be sure to
add point P to the center.
3. Without moving the triangle, use the ‘Distance or Length’ tool to find the distance
between the Incenter and each segment.
P to AB = _______
P to BC = _______
P to CA = _______
4. Pick any vertex of your triangle and drag to change the shape of your triangle. Observe how the
incenter moves.
a. Where is the incenter located when the triangle is acute? (Inside the triangle, outside the
triangle, etc…)
____________________________________
b. … when the triangle is obtuse?
____________________________________
c. … when the triangle is right?
____________________________________
5. Using the numbers you discovered in question 3 to complete the theorem: The incenter of a triangle
is ____________ from the _____________ of a triangle.
6. IMPORTANT! Use the ‘circle with 3 points’ tool to draw a circle inside the triangle that touches all
three sides of the triangle. Where is the incenter in relation to the circle?
______________________________________________________________
Part 3: Orthocenters
1. The orthocenter is the point of concurrency of the ________________________ of a triangle.
2. Erase your drawing from Part 2 and start by drawing a new triangle. Construct its orthocenter.
3. Pick any vertex of your triangle and drag to change the shape of your triangle. Observe how the
circumcenter moves.
a. Where is the incenter located when the triangle is acute? (Inside the triangle, outside the
triangle, etc…)
____________________________________
b. … when the triangle is obtuse?
____________________________________
c. … when the triangle is right?
____________________________________
Part 4: Centroids
1. The centroid is the point of concurrency of the ________________________ of a triangle.
2. Erase your drawing from Part 3 and start by drawing a new triangle. Construct its
centroid. This construction is slightly more complicated, so refer to the picture at
the right. **Be sure your lettering is correct! If GeoGebra does not choose
the correct letter for you, right click on the letter and select ‘Rename.’
Enter the correct letter.**
3. Be sure to add point P to the center. Without moving the triangle, use the ‘Distance or Length’ tool
to find the lengths of each of the segments.
AP = _______
BP = _______
CP = _______
AY = _______
BZ = _______
CX = _______
4. Using the information above, find the relationship between each length. Fill in the blank with a number.
AP = ______ AY
BP = ______ BZ
CP = ______ CX
5. Using the numbers you discovered in question 4 to complete the theorem: The centroid of a triangle
is located ____________ of the distance from each vertex to the midpoint of the opposite side.