Explain cevian fomula.......
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What is the Cevian Triangle formula?
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Girija Warrier, Passionate about Maths & Music...
Answered Mar 25 2018 · Author has 2.1kanswers and 1.8m answer views

CEVIAN is a line segment in a triangle, joining any vertex of the triangle with any one point on its opposite side, other than the end points of the opposite side.
Like in the above image, AD is a Cevian. Yes! All medians, altitudes, angle bisectors of any triangle can be referred as Cevians. But with certain extra characteristics.
TO FIND CEVIAN TRIANGLE FORMULA:
In the above image, angle theeta' ( representing here by (β') is the supplement of angle theeta (representing here by (β).
So, cos β' = -cosβ
Now, by cosine formula…
c² = m² + d² - 2 m d cosβ ………… (1)
b² = n² + d² - 2 n d cos β'
Or, b² = n² + d² + 2 n d cosβ …………..(2)
Eq(1) * n & Eq(2) * m
nc² = nm² + nd² - 2mn d cosβ
mb² = mn² + md² +2mn d cosβ
By adding the above 2 equations, we get…
nc² + mb² = mn ( m+ n) + d² (m + n)
=> mb² + nc² = ( m+ n) ( mn + d²)
=> mb² + nc² = a ( mn + d² ) . . . . Cevian triangle formula
The above relation is Cevian triangle formula or Stewart's Theorem.
With the help of which , we may calculate the length of Cevian (d)
For example, if we need to calculate the length of median cevian. We may use the above formula by taking m = n & m+ n = a
b² m + c² n = a ( d² + mn) ( cevian formula)
For median cevian
b² m + c² m = 2m ( d² + m² )
=> m ( b² + c² ) = 2m ( d² + m² )
=> b² + c² = 2 ( d² + m² )
By multiplying LHS & RHS by 2
=> 2( b² + c² ) = 4 ( d² + m²)
=> 2b² + 2c² = 4d² + (2m)²
=> 2b² + 2c² = 4d² + a²
=> d² = (2b² + 2c² - a²)/4
=> median cevian = √(2b² + 2c² - a²) / 2
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Hungenahalli Sitaramarao Badarinath, Ph.D. in Civil Engineering. Maths keeps one mentally active.
Answered Jun 2 2017 · Author has 13.7kanswers and 4.6m answer views
Consider a triangle ABC, <A at the top, CB as the base, <C on the left, <B on the right. Let D be a point on CB.
Let CB = a,
CA = b,
AB = c,
AD = d,
CD = n, and
DB = m.
Here AD = d is called the cevian.
The cevian length, d, based on Stewart’s theorem is given by
d = [b^2*m + c^2*n - amn]^0.5/a
where a, b, c, m, n are sides and
a = m+n.
Still have a question? Ask your own!
What is your question?
2 ANSWERS

Girija Warrier, Passionate about Maths & Music...
Answered Mar 25 2018 · Author has 2.1kanswers and 1.8m answer views

CEVIAN is a line segment in a triangle, joining any vertex of the triangle with any one point on its opposite side, other than the end points of the opposite side.
Like in the above image, AD is a Cevian. Yes! All medians, altitudes, angle bisectors of any triangle can be referred as Cevians. But with certain extra characteristics.
TO FIND CEVIAN TRIANGLE FORMULA:
In the above image, angle theeta' ( representing here by (β') is the supplement of angle theeta (representing here by (β).
So, cos β' = -cosβ
Now, by cosine formula…
c² = m² + d² - 2 m d cosβ ………… (1)
b² = n² + d² - 2 n d cos β'
Or, b² = n² + d² + 2 n d cosβ …………..(2)
Eq(1) * n & Eq(2) * m
nc² = nm² + nd² - 2mn d cosβ
mb² = mn² + md² +2mn d cosβ
By adding the above 2 equations, we get…
nc² + mb² = mn ( m+ n) + d² (m + n)
=> mb² + nc² = ( m+ n) ( mn + d²)
=> mb² + nc² = a ( mn + d² ) . . . . Cevian triangle formula
The above relation is Cevian triangle formula or Stewart's Theorem.
With the help of which , we may calculate the length of Cevian (d)
For example, if we need to calculate the length of median cevian. We may use the above formula by taking m = n & m+ n = a
b² m + c² n = a ( d² + mn) ( cevian formula)
For median cevian
b² m + c² m = 2m ( d² + m² )
=> m ( b² + c² ) = 2m ( d² + m² )
=> b² + c² = 2 ( d² + m² )
By multiplying LHS & RHS by 2
=> 2( b² + c² ) = 4 ( d² + m²)
=> 2b² + 2c² = 4d² + (2m)²
=> 2b² + 2c² = 4d² + a²
=> d² = (2b² + 2c² - a²)/4
=> median cevian = √(2b² + 2c² - a²) / 2
646 Views · View 2 Upvoters
Sponsored by ExpertOption
Try trading today with a free ExpertOption demo account.
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RELATED QUESTIONS (MORE ANSWERS BELOW)
What is the general formula to find the total number of triangles in Nth Sierpinski's triangle?
401 Views
What is the area of an isosceles triangle without using Heron’s formula or b×h/3?
534 Views
How many triangles are in this picture?
585,469 Views
What is the altitude 's formula of an isosceles triangle?
257 Views
What is the formula to calculate scalane triangle?
383 Views
IN OTHER LANGUAGES
En español: ¿Cuál es la fórmula del triángulo de Cevian?
En français : Qu'est-ce que la formule du triangle cevien ?
In italiano: Qual è la formula del triangolo Cevian?
OTHER ANSWERS

Hungenahalli Sitaramarao Badarinath, Ph.D. in Civil Engineering. Maths keeps one mentally active.
Answered Jun 2 2017 · Author has 13.7kanswers and 4.6m answer views
Consider a triangle ABC, <A at the top, CB as the base, <C on the left, <B on the right. Let D be a point on CB.
Let CB = a,
CA = b,
AB = c,
AD = d,
CD = n, and
DB = m.
Here AD = d is called the cevian.
The cevian length, d, based on Stewart’s theorem is given by
d = [b^2*m + c^2*n - amn]^0.5/a
where a, b, c, m, n are sides and
a = m+n.
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