Cevian Calculator
Calculate medians, altitudes, and angle bisectors of a triangle
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Understanding Cevians
A cevian is any line segment joining a vertex of a triangle to a point on the opposite side. The three most important types of cevians are medians, altitudes, and angle bisectors.
Median Formulas
A median connects a vertex to the midpoint of the opposite side. All three medians intersect at the centroid, which divides each median in a 2:1 ratio.
Altitude Formulas
An altitude is perpendicular to the opposite side. All three altitudes intersect at the orthocenter.
Angle Bisector Formulas
An angle bisector divides the vertex angle into two equal angles. All three angle bisectors intersect at the incenter, the center of the inscribed circle.
Triangle Centers
- Centroid: Intersection of medians (center of mass)
- Orthocenter: Intersection of altitudes
- Incenter: Intersection of angle bisectors (center of incircle)
- Circumcenter: Intersection of perpendicular bisectors (center of circumcircle)
Frequently Asked Questions
What is the difference between a median and an altitude?
A median connects a vertex to the midpoint of the opposite side, while an altitude is perpendicular to the opposite side. They are only the same in isosceles triangles (when drawn from the vertex between the equal sides).
Where do the three medians meet?
The three medians always intersect at a single point called the centroid. This is the center of mass of the triangle and divides each median in a 2:1 ratio from vertex to midpoint.
Can altitudes be outside the triangle?
Yes! In obtuse triangles, two of the three altitudes lie outside the triangle. You need to extend the sides to find where the perpendiculars meet them.
What does the angle bisector theorem state?
The angle bisector theorem states that the angle bisector divides the opposite side in the ratio of the adjacent sides. If the bisector from A meets BC at D, then BD/DC = AB/AC.
How are cevians used in real life?
Cevians are used in structural engineering (finding centers of mass), surveying (triangulation), computer graphics (mesh subdivision), and physics (calculating moments and equilibrium points).
What is special about equilateral triangles?
In equilateral triangles, all medians, altitudes, and angle bisectors from each vertex coincide (they're the same line). All four triangle centers (centroid, orthocenter, incenter, circumcenter) are at the same point.
Can I use these formulas for any triangle?
Yes, these formulas work for all triangles (scalene, isosceles, equilateral, acute, right, obtuse) as long as the three sides satisfy the triangle inequality theorem.
What is Apollonius's theorem?
Apollonius's theorem relates the median length to the sides: in a triangle with median m_a, we have b² + c² = 2m_a² + a²/2. This provides an alternative derivation of the median formula.