Wolfram Cloud Document

TXM28: Sum of the algebraic distances from any arbitrary point to triangle’s sides

1

.

Caption

Let(O,R),(I,r)bethecircumcircleandincircleofΔABC,respectively.Let

d

M

bethesumofthealgebraicdistancesfromanyarbitrarypointMtothesidesofΔABC.Thenwehave:

d

M

=

2

MI

-

2

MO

2R

+

R

2

+2r

2

.

Code

In[]:=

ManipulateModule{circumsphere,insphere,Mtocentroid,R,r,crossAB,crossBC,crossAC,altAB,altBC,altAC,Mdist,MO,MI},circumsphere=Circumsphere[{AA,BB,CC}];insphere=Insphere[{AA,BB,CC}];Mtocentroid=Line[{M,Mean[{AA,BB,CC}]}];R=circumsphere[[2]];r=insphere[[2]];crossAB=If[RegionIntersection[Mtocentroid,InfiniteLine[{AA,BB}]]===EmptyRegion[2],1,-1];crossBC=If[RegionIntersection[Mtocentroid,InfiniteLine[{BB,CC}]]===EmptyRegion[2],1,-1];crossAC=If[RegionIntersection[Mtocentroid,InfiniteLine[{AA,CC}]]===EmptyRegion[2],1,-1];altAB=TriangleConstruct[{AA,M,BB},"Altitude"];altBC=TriangleConstruct[{BB,M,CC},"Altitude"];altAC=TriangleConstruct[{AA,M,CC},"Altitude"];Mdist=EuclideanDistance@@First[#]&/@{altAB,altBC,altAC};MO=EuclideanDistance[M,circumsphere[[1]]];MI=EuclideanDistance[M,insphere[[1]]];ColumnText@Grid[{{Style["R",Green],Style["r",Red],Style["MO",Green],Style["MI",Red],"

d

c

","

d

a

","

d

b

"},{R,r,MO,MI,crossABMdist[[1]],crossBCMdist[[2]],crossACMdist[[3]]}},Frame->All],Row[{"

d

a

+

d

b

+

d

c

"," = ",Mdist.{crossAB,crossBC,crossAC}}],Row

Row[{Superscript["MI",2],"-",Superscript["MO",2]}]

"2 R"

+

"R"

"2"

+"2 r"," = ",

MI^2-MO^2

2R

+

R

2

+2r,{Graphics[{PointSize[.02],RGBColor[.25,.43,.82],InfiniteLine/@Subsets[{AA,BB,CC},{2}],Blue,{altAB,altBC,altAC},Green,circumsphere,Line[{M,circumsphere[[1]]}],Red,insphere,Line[{M,insphere[[1]]}],Black,Text[Style["A",20,"Label"],AA+{0,.4}],Text[Style["B",20,"Label"],BB+{0,.4}],Text[Style["C",20,"Label"],CC+{0,.4}],Text[Style["M",20,"Label"],M+{0,.4}],Text[Style["O",20,"Label",Green],circumsphere[[1]]+{0,.4}],Text[Style["I",20,"Label",Red],insphere[[1]]+{0,.4}]},ImageSize{450,375},PlotRange{{-4.5,4.5},{-4.5,4.5}},PlotRangePadding.25]},Alignment{Center,Top},{{AA,{-3.`,-2.25`}},{-4,-4},{4,4},Locator},{{BB,{-2.705`,2.685`}},{-4,-4},{4,4},Locator},{{CC,{3.`,-0.75`}},{-4,-4},{4,4},Locator},{{M,{2,2}},{-4,-4},{4,4},Locator}

Out[]=

R	r	MO	MI	d c	d a	d b
3.87051	1.72577	2.71567	3.78856	4.73749	-1.6257	3.17647

d

a

+

d

b

+

d

c

= 6.28826

2 r+

R

2

+

2

MI

-

2

MO

2 R

= 6.28826





3

.

Proof

This is

the solution of Kousik Sett

(a friend from India of the author). My original solution is shorter, however it is derived from my other formula

TXM5

4

.