Cevian

The length of a cevian can be determined by Stewart's theorem: in the diagram, the cevian length d is given by the formula

${\displaystyle \,b^{2}m+c^{2}n=a(d^{2}+mn).}$

Less commonly, this is also represented by the mnemonic

${\displaystyle \,man+dad=bmb+cnc.}$[4]

If the cevian happens to be a median (thus bisecting a side), its length can be determined from the formula

${\displaystyle \,m(b^{2}+c^{2})=a(d^{2}+m^{2})}$

or

${\displaystyle \,2(b^{2}+c^{2})=4d^{2}+a^{2}}$

since

${\displaystyle \,a=2m.}$

Hence in this case

${\displaystyle d={\sqrt {\frac {2b^{2}+2c^{2}-a^{2}}{4}}}.}$

If the cevian happens to be an angle bisector, its length obeys the formulas

${\displaystyle \,(b+c)^{2}=a^{2}\left({\frac {d^{2}}{mn}}+1\right),}$

and[5]

${\displaystyle d^{2}+mn=bc}$

and

${\displaystyle d={\frac {2{\sqrt {bcs(s-a)}}}{b+c}}}$

where the semiperimeter s = (a+b+c)/2.

The side of length a is divided in the proportion b:c.

If the cevian happens to be an altitude and thus perpendicular to a side, its length obeys the formulas

${\displaystyle \,d^{2}=b^{2}-n^{2}=c^{2}-m^{2}}$

and

${\displaystyle d={\frac {2{\sqrt {s(s-a)(s-b)(s-c)}}}{a}},}$

where the semiperimeter s = (a+b+c) / 2.