SLC S23 Week2 || Geometry with GeoGebra: The Triangle and Its Elements

Hello everyone, looks like I am back from my holiday just in time to take part in the 2nd week of @sergeyk's geometry class, which can be found here SLC S23 Week2 || Geometry with GeoGebra: The △riangle and Its Elements△. Thank you @ady-was-here for the invitation!

Now let's get started!

_{Image taken from Pixabay}

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.
‼️You must construct and highlight all three altitudes of the triangle.

I started this task with creating a triangle, drawing the 3 altitudes.

_{Starting point}

Next I marked the intersection point with each side, added the segments and the intersection point of the altitudes.

_{Midway through}

From this point, it took a bit of work to find how to achieve the desired result, but after some trial and error it was achieved. Added a bit of styling and decoration and this is the final result:

_{Live representation}

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.

• Show that the altitude is a perpendicular line.
  The altitude is the segment drawn from a vertex perpendicular to the opposite side, representing the shortest distance.
  

  
  
  _Altitude
  

• Show that the median divides the opposite side in half.
  The median represents the segment that connects a vertex to the center of the opposite side, creating 2 smaller triangles with equal area.

_Median

• Show that the angle bisector divides the angle into two equal parts.
  The angle bisector is a segment that divides an angle in two equal, smaller angles.
  
  
  _Bisector
  

Now let's see how the triangle looks with all the lines drawn:

_{Live representation}

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.)
‼️You must construct and show the median triangle – the triangle formed by the midpoints of the sides.

To start this task, we first have to construct a triangle with its 3 medians.

_{Initial construction}

Next, use the Polygon tool to create the triangle that has M¹, M², M³ as vertices and hide the original triangle:

_{Midway through}

After some decoration of the triangles, this is the final result:

_{Live representation}

Some of the key properties of this type of triangle are:

• it has the same angles as the original triangle
• its sides are half the length of the corresponding side of the big triangle
• the 2 triangles share the same center of mass

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.

_{Initial construction}

Next, for me it's easier to use the Polygon tool and just select the 3 points or feet of the altitudes to create the orthic triangle. Below you can see a live representation:

_{Live representation}

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.

For this task, I first started with a simple triangle for which I added the angle bisectors, added the intersection point and marked the angles:

_{Initial construction}

Next, I added the point of intersection between the bisector and each side of the triangle, created the inside triangle and did some styling to look better.

_{Live representation}

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 final task is a combo of all the other ones, in a single triangle. As always, the starting point is a triangle with all the altitudes, medians and bisectors drawn and marked.

_{All lines drawn}

Now that we have everything drawn, we just have to construct each triangle. I hid the parts just to be easier to see the 3 triangles:

_{Live representation}

As always, thank you for reading and I'd like to extend an invitation to @radudangratian to take part in this.
See you next time!

_{Posted using SteemPro}