Special segments in a triangle
Of all the segments you could draw inside a triangle, which ones are worth a name, and what does each one know?
Where you are 4 inside
What this area is about
A few segments in a triangle recur everywhere: ones that halve a side, halve an angle, or meet a side at a right angle. Each carries a fact about lengths, distances or parallels.
How these fit together
Definitions. Medians and altitudes: a median runs from a vertex to the midpoint of the opposite side; an altitude runs from a vertex perpendicular to the line containing the opposite side, so in an obtuse triangle two altitudes fall outside it.
Results about equal distances. Perpendicular bisectors of a triangle: in a plane, a point is equidistant from A and B if and only if it lies on the perpendicular bisector of \overline{AB}, so the three in a triangle meet at one point. Angle bisectors of a triangle: a point inside an angle is equidistant from its two sides if and only if it lies on the bisector, so a triangle’s three angle bisectors meet at one point too.
A result about parallels. Midsegments of triangles and trapezoids: in a triangle, the segment joining the midpoints of two sides is parallel to the third side and half as long; in a trapezoid, the segment joining the midpoints of the legs is parallel to the bases and its length is their average.
A path through it
Medians and altitudes come before midsegments. Perpendicular bisectors and angle bisectors fit anywhere.
The points where each family of three meets are the subject of triangle centers. Also next: the triangle angle bisector theorem, geometric means in a right triangle and coordinate proofs.