SLC S23 Week2 || Geometry with GeoGebra: The Triangle and Its Elements
Hello! You just intersected my day! Today I am posting the homework of @sergeyk's geometry class that you can find here SLC S23 Week2 || Geometry with GeoGebra: The △riangle and Its Elements△.
Image taken from Pexels
Let's get started.
Build a triangle with three altitudes.
When constructing the three altitudes, the point of intersection of the altitudes should always be displayed, as in task #5 (from the previous lesson), where the extensions of the altitudes to their intersection should be shown.
The animation shows half of the task completed. It should be done as shown in the image below. The altitudes must be displayed and also extended to the point of intersection.
‼️You must construct and highlight all three altitudes of the triangle.
Started the task with the creation of the triangle ABC from which I took each Vertex individually and drew a perpendicular line to the opposite side to create one altitude for each one. I added a point of intersection between each altitude and the triangle's side and also one at the intersection of the altitudes.
From this point I added segments between the points and hidden the lines created by the perpendiculars because these were going outside the triangle area, since perpendiculars helped me get the exact points replacing them with segments will do exactly the same thing. Once done I added some stylization to show distinction between the lines, the result looks like this.
Build a triangle. From vertex A, construct (show) the altitude, angle bisector, and median.
Show that the altitude is a perpendicular line.
Show that the median divides the opposite side in half.
Show that the angle bisector divides the angle into two equal parts.
Make sure the altitude, median, and angle bisector stand out to draw attention to them.
‼️You must construct and show one altitude, one median, and one angle bisector from the same vertex.
Altitude represents the shortest path from a Vertex to the opposite side, also known as height, to prove a line is an altitude it needs to create a 90 degree angle with the opposite side.
Median represents a segment that connects a Vertex to the middle of the opposite side, using a median we create two smaller triangles with equal area. To prove my line is a median I showed that the point on the opposite side splits the side in two equal parts.
Bisector is the segment that cuts an angle in two smaller equal angles, in my case I showed that the new angles created by my bisector segment will always have the same value.
Below you can see each line individually without anything added.
And here, you can see a GIF with a live representation, you can see each details for each mentioned segment above:
- you can see the 90 degree angle doesn't change meaning the altitude is correct, always showing the shortest path to the opposite side
- the segments AF and FC share the same size, meaning the F is the middle of that size and the segment BF is a median
- for the bisector I added the two smaller angles with values and you can see no matter how I change the shape of the triangle, the angles have the same value meaning our segment is a bisector
The Basics of Medians
Build triangle ABC. Then, using the medians as the new triangle's vertices, construct a new triangle.
This triangle has several names: Medivan triangle, Ceva’s triangle, and Median triangle.
Construct triangle ABC, then build a new triangle M1M2M3 using the midpoints of the medians as the vertices.
Hide the sides of triangle ABC, but display the halves of the sides and demonstrate that they are equal. (Use the Style/Decoration option to adjust the appearance.)
What properties does this triangle have?
‼️You must construct and show the median triangle – the triangle formed by the midpoints of the sides.
I started this task by creating my ABC triangle and after that I used Midpoint to find and create a point in the middle of each one of the sides.
I named each point accordingly (M_1, M_2, M_3), also I added a measurement for each half of the side to prove the equality between them and prove I am working with Medians.
After that I used to connect the newly created points (M_1, M_2, M_3) with segments to create the Medivan triangle.
Below you can also check two GIFs with two states of the triangles.
| GIF | GIF |
|---|---|
 |  |
Now what are some key properties for the Median Triangle:
- Each side has half the length of the original triangle side
Example:
The Median Triangle has the same shape as the original triangle but it is rotated and scaled down
The area of the Median Triangle is one quarter of the original triangle (1/4)
The Median triangle is an acute one no matter how you move/rotate it, meaning all it's inside angles are below 90 degrees
If we draw the medians of the Median Triangle these will intersect in the same point as the original Triangle
The Bases of the Altitudes
Build triangle ABC, then, using the bases of the altitudes as the vertices, construct another triangle.
‼️You must construct and show the orthic triangle – the triangle formed by the feet of the altitudes.
Again started by creating a random ABC triangle, after that I created an altitude (height) from each vertex to the opposite side, on the opposite side I added a point at the intersection of the altitude with the side, because the perpendicular tool creates a line that goes past the triangle I replaced them with segments and marked their intersection. To match @sergeyk's example I also added the 90 degree angles for each altitude.
Now with everything ready ready all I had left to do is to connect the altitude points to create a new triangle.
The result looks like this:
The Bases of the Angle Bisectors
Construct triangle ABC, then, using the bases of the angle bisectors as the vertices, construct another triangle.
Show that the angles formed are exactly the angle bisectors.
‼️You must construct and show the incircle triangle – the triangle formed by the feet of the angle bisectors.
If in the previous task we created a triangle out of the intersection between altitudes and sides (point of intersection), this time we are going to use bisectors.
I created my triangle and a bisector from each angle, at the intersection of the bisector with the opposite side I added a point. Because the bisectors line goes beyond the triangle I replaced them with segments to keep everything inside the triangle. I also marked each angle to prove they share the same value and they are split in even values.
And a live representation here:
Display Four Triangles Together
Draw the four triangles: the main triangle ABC, and the triangles formed by the bases of the altitudes, angle bisectors, and medians.
There should be four (or three) triangles on the drawing. (It is normal for the triangle formed by the bases of the altitudes to disappear.)
‼️You must construct and show four triangles: the main triangle and the three triangles formed by the altitudes, medians, and angle bisectors.
Note: The orthic triangle may not always be visible.
This task is a combination of the ones above, I created the main triangle and drew all the medians, bisectors and altitudes, marked their specific elements:
- 90 degree angles for altitudes
- equal angles for bisectors
- equal length for sides split by medians
Now there aren't many things to explain here, since each triangle has been done individually in the previous task, I am going to show you two live representations one with all the elements and the other with a few of them turned off so you can understand everything easier.
| Full GIF | Hidden Parts GIF |
|---|---|
 |  |
That was it for this homework, the difficulty increased a little from the previous homework, I'd say Task 1 was the one that was harder to create, it took me some tries before I understood what needs to be done (hopefully I understood it as it should). I tried to keep it short and on point without spamming too many pictures. In the end I'd like to invite @mojociocio to give it a try.