lpetrich
Contributor
Triangles have a Magic Highway - Numberphile - YouTube
A triangle's Euler line.
Medicenter:
Send a line from each vertex to the median / center point on the opposite side.
All three lines will intersect at one point, the medicenter, given by the average of the three vertices' positions. This is also the barycenter of the triangle if it has constant surface density, the center of mass or the centroid.
Orthocenter:
Send a line from each vertex to the opposite side, such that the line is perpendicular to each side.
All three lines will intersect at one point, the orthocenter.
Circumcenter:
Center of the circumscribed circle, the circle that passes through the triangle's vertices.
Incenter:
Center of the inscribed circle, the circle that touches the edges without crossing them.
Euler line mentions some more triangle-related points that lie it. The medicenter, orthocenter, and circumcenter are on it, but for a scalene triangle (all side lengths unequal), the incenter is not on it.
For an isosceles triangle (two side lengths equal), the Euler line is the line of symmetry, and the incenter is also on it
For an equilateral triangle (all side lengths equal), all these points coincide.
Triangle Centres and the Euler Line (extra footage) - YouTube
Prof. Zvezdelina Stankova, who presented these results, is from Bulgaria.
A triangle's Euler line.
Medicenter:
Send a line from each vertex to the median / center point on the opposite side.
All three lines will intersect at one point, the medicenter, given by the average of the three vertices' positions. This is also the barycenter of the triangle if it has constant surface density, the center of mass or the centroid.
Orthocenter:
Send a line from each vertex to the opposite side, such that the line is perpendicular to each side.
All three lines will intersect at one point, the orthocenter.
Circumcenter:
Center of the circumscribed circle, the circle that passes through the triangle's vertices.
Incenter:
Center of the inscribed circle, the circle that touches the edges without crossing them.
Euler line mentions some more triangle-related points that lie it. The medicenter, orthocenter, and circumcenter are on it, but for a scalene triangle (all side lengths unequal), the incenter is not on it.
For an isosceles triangle (two side lengths equal), the Euler line is the line of symmetry, and the incenter is also on it
For an equilateral triangle (all side lengths equal), all these points coincide.
Triangle Centres and the Euler Line (extra footage) - YouTube
Prof. Zvezdelina Stankova, who presented these results, is from Bulgaria.