Stewart's Theorem Calculator
Calculate cevian length using Stewart's theorem
Triangle Parameters
Results
Understanding Stewart's Theorem
Stewart's theorem relates the length of a cevian (a line segment from a vertex to the opposite side) to the sides of the triangle and the segments created on the side. It's named after Scottish mathematician Matthew Stewart (1717-1785).
The Theorem
Variables
- a - The side BC (= m + n)
- b - The side AC
- c - The side AB
- d - The cevian AD
- m - The segment BD
- n - The segment DC
Solving for Cevian Length
Rearranging Stewart's theorem to solve for d:
d = √((b²m + c²n - amn) / a)
Special Cases
- Median: When m = n = a/2, the cevian is a median
- Altitude: Special case when d is perpendicular to BC
- Angle bisector: When d bisects angle A
Mnemonic
"A man and his dad put a bomb in the sink" - helps remember: b²m + c²n = a(d² + mn)
Frequently Asked Questions
What is a cevian?
A cevian is a line segment that joins a vertex of a triangle to a point on the opposite side (or its extension). Medians, altitudes, and angle bisectors are all special types of cevians.
Who was Matthew Stewart?
Matthew Stewart (1717-1785) was a Scottish mathematician who published this theorem in 1746. However, the theorem may have been known earlier to other mathematicians.
When should I use Stewart's theorem?
Use Stewart's theorem when you need to find the length of a cevian and you know the three sides of the triangle and how the cevian divides the opposite side. It's particularly useful for finding median lengths.
Can I use it for altitudes?
Yes, but for altitudes there are often simpler methods. Stewart's theorem works for any cevian, including altitudes, but you may prefer using trigonometry or the Pythagorean theorem for right triangles.
How do I find median length?
For a median, m = n = a/2. Substitute into Stewart's theorem to get the median length formula: d² = (2b² + 2c² - a²) / 4, which simplifies to d = ½√(2b² + 2c² - a²).
What if I get a negative d²?
If d² is negative, your input values don't form a valid triangle or the segments m and n don't correctly divide side a. Check that the triangle inequality holds and m + n = a.
Is there a proof of Stewart's theorem?
Yes, Stewart's theorem can be proven using the law of cosines applied to the two smaller triangles formed by the cevian, or by using coordinate geometry and algebraic manipulation.
What are practical applications?
Stewart's theorem is used in surveying to calculate distances, in structural engineering for truss analysis, in computer graphics for mesh subdivision, and in competition mathematics.