Triangle Properties
Here are visuals for each theorem:
Angle Bisector Theorem:
- \(\dfrac{\overline{CA}}{\overline{CD}}=\dfrac{\overline{BA}}{\overline{DB}}\)
Median Theorem of a Triangle:
Concurrency and medians theorem
The distance from a vertex to the centroid is two-thirds the length of the median.
If \(\overline{AF}\), \(\overline{BE}\) and \(\overline{CD}\) are medians, then \(AG=\dfrac{2}{3} AF\), \(BG=\dfrac{2}{3}BE\) and \(CG=\dfrac{2}{3} CD\).
Perpendicular Bisector Theorem:
Theorem: Concurrency of Perpendicular Bisectors of a Triangle
- The perpendicular bisectors of a triangle intersect at a point that is equidistant from the vertices of a triangle.
- \(PA=PB=PC\)
Altitude of a Triangle:
An altitude of a triangle is a perpendicular segment that joins a vertex of the triangle to the opposite side.
- \(\dfrac{\text{side 1}}{\color{red}{altitude}}=\dfrac{\color{red}{altitude}}{\text{side 2}}\)
