Remarkable Lines in a Triangle - Part 1

追加: 2026-05-14

[00:00:09]Hello dear students, how are you? Today we are going to learn about remarkable lines and triangles. What do we mean by remarkable lines? Remarkable lines and triangles might be heights, medians, bis sectors of the angles or perpendicular bis sectors of the segments the sides of the uh triangles. So let's see together.

[00:00:39]First let's start with the perpendicular bis sectors. You know that we draw we use the compass to draw the perpendicular bis sector of the segments. Here I've drawn two uh perpendicular bis sectors and I'm going to draw the third one the perpendicular bis sector of AC. So to draw the perpendicular bis sector I place it on the first uh on the first extremity of the segment. I drew two arcs, one up, one down. You know that we've learned this.

[00:01:17]And on the second one up and down. Then the two uh the two points of intersection of the two arcs should be joined together.

[00:01:34]We join them together. Then we will have the perpendicular bis sector.

[00:01:48]I've already drawn the two the two other perpendicular bis sectors. The perpendicular bis sector of BA and the perpendicular bis sector of B uh C. We know that the perpendicular bis sector uh divides the segment into two equal parts and it is uh it intersects the segment with a 90° angle. So now we realize or we recognize that the three perpendicular bis sectors of this triangle they meet at one point. They meet at the same point. This point is called the the point of uh or the center of the circle circumscribed uh uh by the triangle. This means that or circum circumscribed about the triangle. This means that if I open the compass and put the the needle on this point look on this point exactly. And I open for example on A. If I draw the circle, the circle will be passing through the three points.

[00:03:10]It is passing through the point C, passing through the point A and passing through the point B. A, B, C. These three points are on the circumference of the circle. Because of this we say that this circle of this center let's call it O for example and let's call this this circle C this circle C of center O circumscribe about the triangle A B C now let's speak about the heights the height is the perpendicular line that is originated from the vertex of the uh uh from the triangle to its opposite side. We use the set square to do it. So this is this is the first one.

[00:03:59]Move. Here it is.

[00:04:11]This is the first one.

[00:04:14]Perpendicular. Now from point B to the line AC, we draw a perpendicular also.

[00:04:21]Here it is.

[00:04:34]And from the point C, we have to draw a perpendicular to the line AB.

[00:04:41]Place the set square on AB and the second edge passing through C and draw The perpendicular as you see also in this case the three lines intersect in the same point. This point is called the ortho center of the triangle A B C. So let's call this point N. So n is the ortho center.

[00:05:23]Now we are going to speak about the medians. The median is the uh is the line that comes or is originated from the vertex to the midpoint of this of the uh side that faces this uh point.

[00:05:41]For example, here the uh the side that faces the point C is AB. I've already uh put or located the midpoints of the three lines. P is the midpoint of AB. N is the midpoint of AC and M is the midpoint of B C. Now let's join the three vertices of the triangle to the three midpoints.

[00:06:24]As we see that the three uh midpoints or the three lines they intersect in the same uh point which is here. Let us call it the point uh O for example. O is the point of intersection of the three medians of the triangle. So O is called the centroid.

[00:06:51]Centr is the point of intersection of the medians.

[00:07:04]Again, the medians are the lines that are originated from the vertices of the triangle to the midpoints of the sides of the triangle.