Cevian

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Definition

A cevian is a line segment or ray that extends from one vertex of a polygon (usually a triangle) to the opposite side (or the extension of that side). In the below diagram, $AD$ is a cevian.

[asy] draw((0,0)--(100,0)--(10,50)--(0,0)); draw((10,50)--(70,0)); dot((10,50)); label("$A$",(10,50),N); dot((0,0)); label("$B$",(0,0),SW); dot((100,0)); label("$C$",(100,0),SE); dot((70,0)); label("$D$",(70,0),S); [/asy]

Special Cevians

  • A median is a cevian that divides the opposite side into two congruent lengths.
  • An altitude is a cevian that is perpendicular to the opposite side.
  • An angle bisector is a cevian that divides the angle the cevian came from in half.

Finding Lengths

[asy] draw((0,0)--(100,0)--(10,50)--(0,0)); draw((10,50)--(70,0)); dot((10,50)); label("$A$",(10,50),N); dot((0,0)); label("$B$",(0,0),SW); dot((100,0)); label("$C$",(100,0),SE); dot((70,0)); label("$D$",(70,0),S); [/asy]

In the diagram, note that $\cos{ \angle ADB} = -\cos{ \angle ADC}$ because $\angle ADB + \angle ADC = 180^\circ$. Thus, \[\frac{AD^2 + DB^2 - AB^2}{2 \cdot AD \cdot DB} = -\frac{AD^2 + DC^2 - AC^2}{2 \cdot AD \cdot DC}\]

See also

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