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Triangle Cevian Lengths

tricev.html

Exploration

Prove that the length of the altitude, "tricev_1.gif", from vertex "tricev_2.gif" of a triangle to the opposite side of the triangle (whose length is "tricev_3.gif") is given by

"tricev_4.gif"

where "tricev_5.gif", "tricev_6.gif" and "tricev_7.gif", "tricev_8.gif" and "tricev_9.gif" are the lengths of the triangle's sides.  Prove that the length of the median, "tricev_10.gif", from vertex "tricev_11.gif" is given by

"tricev_12.gif"

Prove that the length of the angle bisector, "tricev_13.gif", from vertex "tricev_14.gif" is given by

"tricev_15.gif".

Also show that the formulas for the lengths of the cevians from vertices "tricev_16.gif" and "tricev_17.gif" can be found by cyclic permutation of the subscripts given in the formulas above.

Approach

Construct a triangle with the given side lengths.  Construct the associated cevians (altitude, median and angle bisector) and compute and simplify the expressions for their lengths.

Initialize

To initialize Descarta2D, select the input cell bracket and press SHIFT-Enter.

This initialization assumes that the Descarta2D software has been copied into one of the standard directories for AddOns which are on the Mathematica search path, $Path.

<<Descarta2D`

Solution

Construct a triangle with the given side lengths and simplify.

Clear[s,s1,s2,s3,S,P,e1];
T=(Triangle2D[{s1,s2,s3}] //FullSimplify) /.
     Sqrt[-e1_/s_Symbol^2]->Sqrt[-e1]/s

"tricev_18.gif"

The altitude of a triangle is the cevian perpendicular to the opposite side.  These functions return the length of the altitude (the height) for each vertex, 1, 2 or 3.  The perpendicular is found by projecting the vertex on the line containing the opposite side.

Height2D[Triangle2D[p1:{x1_,y1_},p2:{x2_,y2_},p3:{x3_,y3_}],
         n_ /; (n==2 || n==3)]:=
   Height2D[Triangle2D[p2,p3,p1],n-1];

Height2D[Triangle2D[p1:{x1_,y1_},p2:{x2_,y2_},p3:{x3_,y3_}],1]:=
   Distance2D[p1,Coordinates2D[Point2D[p1],Line2D[p2,p3]]];

Compute the length of each altitude (the height) using the functions defined above and simplify.

(Map[Height2D[T,#]&, {1,2,3}] //FullSimplify) //.
   {Sqrt[-e1_/s_Symbol^2]->Sqrt[-e1]/s,
    Sqrt[e1_/s_Symbol^2]->Sqrt[e1]/s,
    s1+s2+s3->S,
    (s1-s2-s3)(s1+s2-s3)(s1-s2+s3)->-P,
    (-s1+s2-s3)(s1+s2-s3)(-s1+s2+s3)->-P,
    (s1+s2-s3)(s1-s2+s3)(-s1+s2+s3)->P}

"tricev_19.gif"

The median of a triangle is the cevian connecting a vertex to the midpoint of the opposite side.  These functions return the length of the median for each vertex, 1, 2 or 3.

Median2D[Triangle2D[p1:{x1_,y1_},p2:{x2_,y2_},p3:{x3_,y3_}],
         n_ /; (n==2 || n==3)]:=
   Median2D[Triangle2D[p2,p3,p1],n-1];

Median2D[Triangle2D[p1:{x1_,y1_},p2:{x2_,y2_},p3:{x3_,y3_}],1]:=
   Distance2D[p1,(p2+p3)/2];

Compute the length of each median using the functions defined above and simplify.

Map[Median2D[T,#]&, {1,2,3}] //FullSimplify

"tricev_20.gif"

The angle bisector of a triangle is the cevian bisecting the angle of the vertex.  These functions return the length of the angle bisector for each vertex, 1, 2 or 3.  Note that the angle bisector must pass through the center of the inscribed circle.

Bisector2D[Triangle2D[p1:{x1_,y1_},p2:{x2_,y2_},p3:{x3_,y3_}],
           n_ /; (n==2 || n==3)]:=
   Bisector2D[Triangle2D[p2,p3,p1],n-1];

Bisector2D[T:Triangle2D[p1:{x1_,y1_},p2:{x2_,y2_},p3:{x3_,y3_}],1]:=
   Module[{pt,ln},
      pt=Coordinates2D[Circle2D[T,Inscribed2D]];
      ln=Line2D[p1,pt];
      Distance2D[p1,Coordinates2D[ln,Line2D[p2,p3]]] ];

Compute the length of each angle bisector using the functions defined above and simplify.

ans1=Map[FullSimplify[Bisector2D[T,#],Assumptions->{s1>0,s2>0,s3>0}]&, {1,2,3}]

"tricev_21.gif"

"tricev_22.gif"

"tricev_23.gif"

"tricev_24.gif"

"tricev_25.gif"

"tricev_26.gif"

"tricev_27.gif"


Copyright © 1999-2007 Donald L. Vossler, Descarta2D Publishing
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