Cevian

Length

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

b^{2} m + c^{2} n = a (d^{2} + m n) .

Less commonly, this is also represented (with some rearrangement) by the following mnemonic:

\underset{A m a n and his d a d}{m a n + d a d} = \underset{put a b o m b in the s i n k .}{b m b + c n c}

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

m (b^{2} + c^{2}) = a (d^{2} + m^{2})

2 (b^{2} + c^{2}) = 4 d^{2} + a^{2}

since

a = 2 m .

Hence in this case

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

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

(b + c)^{2} = a^{2} (\frac{d^{2}}{m n} + 1),

and

d^{2} + m n = b c

and

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

where the semiperimeter $s = \frac{a + b + c}{2} .$

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

d^{2} = b^{2} - n^{2} = c^{2} - m^{2}

and

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

where the semiperimeter $s = \frac{a + b + c}{2} .$

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