Stewarts Theorem in Triangle Geometry

• Stewart's theorem in Geometry yields a relation between the cervain length and the side lengths of a triangle. 

• Stewart's Theorem can be proved using the law of cosines as well as by using the famous Pythagorean Theorem. 

• The theorem was proposed in honor of the Scottish mathematician Matthew Stewart in 1746.

Stewart’s Theorem Angle Bisector

In Geometry, an angle bisector is a ray, line, or segment that divides an angle into two equal parts. In a triangle, one angle bisector is a straight line that divides an angle into two equal or congruent angles. Every triangle can have three angle bisectors, one for each vertex. The intersection of these three angle bisectors is the incentre of a triangle. The distance between the incentre and all of a triangle's vertices is the same. According to the angle bisector theorem, in a triangle, the angle bisector drawn from one vertex splits the side on which it falls in the same ratio as the ratio of the triangle's other two sides. 

Stewart’s Theorem Proof:

Theorem statement – A triangle's angle bisector divides the opposing side into two segments that are proportionate to the triangle's other two sides.

Here’s the Stewart’s Theorem proof,

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In △ABC, point D is a point on BC where AB=c, AC=b, BD=u, DC=v, and AD=t.  

Stewart's theorem states that in this triangle, the following equation given below holds:

In Stewart’s Theorem Geometry, Stewart's theorem can be proved by drawing the perpendicular from the vertex of the triangle up to the base and by using the Pythagoras Theorem for writing the distances b, d, c, in terms of altitude. The right and left-hand sides of the equation reduce algebraically to form the same kind of expression.

Before moving on to the proof by Pythagoras Theorem, let’s know what Pythagoras Theorem is!

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The little box in the corner of the triangle denotes the right angle which is equal to 90 degrees.

• The side opposite to the right angle is the longest side of the triangle which is known as the hypotenuse (H).

• The side that is opposite to the angle is known as the opposite (O).

• And the side which lies next to the angle is known as the Adjacent (A).

• According to Pythagoras theorem, 

In a right-angle triangle,

Proof by Pythagoras Theorem:

Let us assume ∠B and ∠C are both acute angles and where u<v in the figure given above. Then we have,

t2 = h2+ x2

b2 = h2+ (v–x)2 = b2u = h2 u + uv2 – 2uvx+ ux2 

c2 = h2+ (u+x)2 = c2u = h2 v+ u2 v+ 2uvx+ vx2 

b2 u+ c2 u =h2u+ h2v+ uv2 + u2v −2uvx +2uvx + vx2+ ux2

= (u +v) (h2+uv+ x2)

= (u+v)( t2+uv)

= a(t2+uv)

Stewart’s Theorem Statement in Stewart theorem Geometry–

If a, b, c are the lengths of the triangle ABC. Let d be the length of the cevian of the side of the length a. Suppose the cevian d divides the side ‘a’ into 2 segments of the length m and n, where m is equal to the adjacent to the side c and whereas n is equal to the adjacent to the side b, then prove that

b2m+c2 n=a(d2+mn)

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Proof:

The theorem can be completed by using the law of cosines.

Let us consider θ to be the angle between the side m and side ‘d’ and lLet θ′ be equal to the angle between the side n and side d. This angle θ′ is usually the addition of θ and cos θ′ = (− cos θ).

The Law of the cosines for the angles θ′ and θ states that-

c2=m2+d2−2dm cos θ

b2=n2+d2−2dn cos θ′

= n2+d2+2dn cos θ

Now, we need to multiply the first equation by n, and the second equation needs to be multiplied by m, and later add the two equations to remove cos θ, 

Now we obtain

b2 m+c2 n

= nm2+n2m+ (m+ n) d2

= (m+n) (mn+d2)

= a(mn+d2) =  b2 m+c2 n

Therefore, this is the required proof.

Now you might think what a cevian is?

In geometry, a cevian can be defined as any line segment in a triangle with one endpoint on a vertex of the triangle and the other endpoint on the opposite side of the triangle. Special cases of cevians include medians, altitudes, and angle bisectors.

Trick to remember the Stewart Theorem in Stewart theorem Geometry:-

To remember this theorem, use this trick,

(man+dad) = (bmb+cnc)

Or a man (product of m,a,n) and his dad( product of a,d, and d) put a bomb ( product of 

b, m, and b) in the sink (cnc).

Uses of Stewart’s Theorem

In Stewart's Theorem Geometry, Stewart's theorem defines the relationship between the lengths of sides of any given triangle as well as the length of the cevian of the triangle.

Problems to be Solved:

Question 1: Prove Stewart’s Theorem using Pythagoras Theorem.

Answer: Proof by Pythagoras Theorem-

Let us assume ∠B and ∠C are both acute angles and where u<v in the figure given above. Then we have,

t2 =h2+ x2

b2 =h2+ (v−x)2 = b2u = h2 u + uv2 – 2uvx+ ux2 

c2 =h2+ (u+x)2 = c2u = h2 v+ u2 v+ 2uvx+ vx2 

b2 u+ c2 u =h2u+ h2v+ uv2 + u2v −2uvx +2uvx + vx2+ ux2

= (u +v) (h2+uv+ x2)

= (u+v)( t2+uv)

= a(t2+uv) 

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Cavian in a Triangle

A cevian is a line in geometry that touches both the vertices of a triangle and the side opposite that vertex. Cevians are special instances of medians and angle bisectors. The term "cevian" is derived from the Italian mathematician Giovanni Ceva, who developed the well-known theorem regarding cevians that carries his name. A Cevian is a line segment that connects a triangle's vertex to a point on the opposite side (or its extension). Ceva's theorem states that three universal Cevians from the three vertices of a triangle must agree. It should be noted that a cevian is not required to pass through the triangle. It can alternatively travel via the opposite side's extension.