Relationships within Triangles
Lessons in this chapter
About Relationships within Triangles
Relationships within Triangles explores the internal structure of a triangle beyond just its angles and sides.
Angles and Sides in Triangles covers the relationship between a triangle's largest angle and its longest side, before Some Special Points in Triangles and Special Points in Triangles - Construction introduce the centroid, circumcenter, incenter, and orthocenter: four distinct points where different special lines of a triangle all meet.
Triangle Mid-Segment Theorem closes the chapter with a segment connecting the midpoints of two sides, proving it's always parallel to the third side and exactly half its length.
The four lessons in this chapter each open as a Geometry worksheet in the page: Angles and Sides in Triangles, Some Special Points in Triangles, Special Points in Triangles - Construction, and Triangle Mid-Segment Theorem. Questions are scored as you answer them, which helps keep the four special points apart, since they are easy to confuse once all of them have been met. Free to try, with no login needed for your first 3.
Frequently asked questions
Angles and Sides in Triangles relates a triangle's longest side to its largest angle and sets out the triangle inequality, which decides whether three lengths can form a triangle at all. Some Special Points in Triangles introduces the centroid, circumcenter, incenter and orthocenter and what each one is the meeting point of, and Special Points in Triangles - Construction builds them with compass and straightedge. Triangle Mid-Segment Theorem shows that the segment joining two midpoints is parallel to the third side and half its length. The chapter sits in the high school Geometry conceptual category (G).
The largest angle is always opposite the longest side, and the smallest angle is always opposite the shortest side. This relationship holds for every triangle, not just special cases.
The centroid is the point where a triangle's three medians (segments from each vertex to the midpoint of the opposite side) all intersect: it's also the triangle's actual center of balance.
The circumcenter is the point where a triangle's three perpendicular bisectors intersect, and it's equidistant from all three vertices: the center of the circle that passes through all three.
The incenter is the point where a triangle's three angle bisectors intersect, and it's equidistant from all three sides: the center of the circle that fits perfectly inside the triangle.
The orthocenter is the point where a triangle's three altitudes (perpendicular segments from each vertex to the opposite side) intersect.