Properties of Triangles

 Length of side - a (BC) Length of side - b (AC) Length of side - c (AB) if a rectangle is inscribed with one side coinciding with base BC and height h, and value of h is Find Out   ↓ Whether triangle is possible from the length of given sides ? If possible, Perimeter (a+b+c) Maximum possible area within above given perimeter (√3)*a2 / 4 where a = perimeter/3 Area i.e  Δ = √[s(s-a)(s-b)(s-c)] where s=(a+b+c)/2 Perimeter remaining constant, if changed to a circle, radius (p/2π) & area of circle(p2/4π) & area of circle / area of Δ Perimeter remaining constant, if changed to a square, area of square (p2/16) area of square / area of Δ If Δ is converted to an arc of radius x whose value is , then  under the arc & radii, area & arc length are & Perimeter remaining same, maximum area(p2/16), radius under maximum area (p/4), arc length under max. area (p/2) , & Angle Between radii : in radian (p/radius  - 2), in degree & abc abc/p abc/Δ Δ / perimeter Δ / maximum possible area cos A & cos (A/2)  [cos A=(b2 + c2- a2 ) / 2bc  and cos (A/2) =  [ s(s-a)/bc ]1/2 ] and cos B & cos (B/2)  [cos B=(a2 + c2- b2 ) / 2ac  and cos (B/2) =   [  s(s-b)/ac ]1/2 ] and cos C & cos (C/2)  [cos C=(a2 + b2- c2 ) / 2ab  and cos (C/2) =   [ s(s-c)/ab ]1/2  ] and sin A & sin (A/2)   [sin A=2*Δ/bc                      and sin (A/2) =   [ (s-c)(s-b)/bc ]1/2 ] and sin B & sin (B/2)   [sin B=2*Δ/ac                       and  sin  (B/2) = [ (s-a)(s-c)/ac ]1/2 ] and sin C & sin (C/2)   [sinC=2*Δ/ab                       and   sin  (C/2) = [ (s-a)(s-b)/ab ]1/2 ] and Angle A in degree Angle B in degree Angle C in degree Angle A in radian Angle B in radian Angle C in radian Radius of circum-circle [Radius of circum-circle= a/2sinA =b/2sinB=c/2sinC = abc/(4*Δ)] Radius of in-circle         [Radius of in-circle                                                      =    Δ  / s     ] Radius of escribed circle opposite to angle A [radius = Δ/(s-a) ] Radius of escribed circle opposite to angle B [radius = Δ/(s-b) ] Radius of escribed circle opposite to angle C [radius = Δ/(s-c) ] Circum-radius of pedal triangle                       [(radius of circum-circle)/2 ] Length of median AD Length of median BE Length of median CF If H is ortho-center, AH [AH=2*circum-radius *cosA] If H is ortho-center, BH [BH=2*circum-radius *cosA] If H is ortho-center, CH [CH=2*circum-radius *cosA] If AH touches BC at K, then HK [HK=2*circum-radius *cosBcosC] If BH touches AC at L, then  HL [HL=2*circum-radius  *cosAcosC] If CH touches AB at M, then HM [HM=2*circum-radius *cosAcosB] Length of perpendicular from A to BC, AK comment on AK(if angle B or angle C >90 degree, AK outside triangle) Length of perpendicular from B to AC,  BL comment on BL(if angle A or angle C >90 degree, BL outside triangle) Length of perpendicular from C to AB, CM comment on CM(if angle B or angle A >90 degree, CM outside triangle) BK / CK  (ratio in which perpendicular AK divides the side BC) (a*a+c*c-b*b) / (a*a+b*b-c*c) / CL / AL   (ratio in which perpendicular AK divides the side BC) (b*b+a*a-c*c) /(b*b+c*c-a*a) / AM / BM (ratio in which perpendicular AK divides the side BC) (c*c+b*b-a*a)/(c*c+a*a-b*b) / **If a semi-circle is constructed with diameter lying on side c such that the semi-circle touches side b & a, then radius of semi-circle, rc is (from sides)=2*area/(a+b) **If a semi-circle is constructed with diameter lying on side b such that the semi-circle touches side c & a, then radius of semi-circle, rb is (from sides)=2*area/(a+c) **If a semi-circle is constructed with diameter lying on side a such that the semi-circle touches side b & c, then radius of semi-circle, ra is (from sides)=2*area/(c+b) a/(b+c-a) + b/(c+a-b) + c/(a+b-c) = X=3+* = 3 + (a/b) +(b/c)+(c/a) The length of the rectangle with base coinciding with BC is Perimeter of rectangle with base coinciding with BC is area  of rectangle with base coinciding with BC is (area of rectangle is maximum when height is AK/2 & it is half of area of triangle) perimeter of rectangle/perimeter of triangle area of rectangle / area of triangle Co-ordinates of A (x1,y1,z1) (x1)--(y1)--(z1) Co-ordinates of B (x2,y2,z2) (x2)--(y2)--(z2) Co-ordinates of C (x3,y3,z3) (x3)--(y3)--(z3) A point with co-ordinates (x4,y4,z4) whether inside the triangle ? (x4)--(y4)--(z4) Find Out   ↓ Whether triangle is possible ? Whether point is inside triangle? Whether the Point is in the plane of the triangle? yes no Sum of 3 triangles the point makes with 3 vertex of the triangle If possible, length of side a length of side b length of side c If possible, Perimeter (a+b+c) Area cos A & cos (A/2)  [cos A=(b2 + c2- a2 ) / 2bc  and cos (A/2) =  [ s(s-a)/bc ]1/2 ] and cos B & cos (B/2)  [cos B=(a2 + c2- b2 ) / 2ac  and cos (B/2) =   [  s(s-b)/ac ]1/2 ] and cos C & cos (C/2)  [cos C=(a2 + b2- c2 ) / 2ab  and cos (C/2) =   [ s(s-c)/ab ]1/2  ] and sin A & sin (A/2)   [sin A=2*Δ/bc                      and sin (A/2) =   [ (s-c)(s-b)/bc ]1/2 ] and sin B & sin (B/2)   [sin B=2*Δ/ac                       and  sin  (B/2) = [ (s-a)(s-c)/ac ]1/2 ] and sin C & sin (C/2)   [sinC=2*Δ/ab                       and   sin  (C/2) = [ (s-a)(s-b)/ab ]1/2 ] and Angle A in degree Angle B in degree Angle C in degree Angle A in radian Angle B in radian Angle C in radian Radius of circum-circle of the triangle Radius of in-circle of the triangle Radius of escribed circle opposite to angle A [radius = Δ/(s-a) ] Radius of escribed circle opposite to angle B [radius = Δ/(s-b) ] Radius of escribed circle opposite to angle C [radius = Δ/(s-c) ] Circum-radius of pedal triangle Length of median AD Length of median BE Length of median CF Co-ordinate of Centroid(mx1,my1,mz1) -- psum  (absolute value of Ax+By+Cz-D) where x=x4,y=y4,z=z4 If H is ortho-center, AH [AH=2*circum-radius *cosA] If H is ortho-center, BH [BH=2*circum-radius *cosA] If H is ortho-center, CH [CH=2*circum-radius *cosA] If AH touches BC at K, then HK [HK=2*circum-radius *cosBcosC] If BH touches AC at L, then  HL [HL=2*circum-radius  *cosAcosC] If CH touches AB at M,then HM [HM=2*circum-radius *cosAcosB] Length of perpendicular from A to BC, AK Length of perpendicular from B to AC,  BL Length of perpendicular from C to AB, CM BK / CK  (ratio in which perpendicular AK divides the side BC) / CL / AL   (ratio in which perpendicular AK divides the side BC) / AM / BM (ratio in which perpendicular AK divides the side BC) / **If a semi-circle is constructed with diameter lying on side c such that the semi-circle touches side b & a, then radius of semi-circle, rc is (from sides)=2*area/(a+b) **If a semi-circle is constructed with diameter lying on side b such that the semi-circle touches side c & a, then radius of semi-circle, rb is (from sides)=2*area/(a+c) **If a semi-circle is constructed with diameter lying on side a such that the semi-circle touches side b & c, then radius of semi-circle, ra is (from sides)=2*area/(c+b) a/(b+c-a) + b/(c+a-b) + c/(a+b-c) = X=3+* = 3 + (a/b) + (b/c) + (c/a) Equn. of plane passing through A, B, C (a2x+b2y+c2z+d2=0) x +y+z = *Equn. of AB in 3-D (a1x+b1y+c1z+d1=0) &  (a2x+b2y+c2z+d2=0) =>ax+by+cz+d=0 where a=(a1-a2), b=(b1-b2),c1=(c1-c2) & d=d1-d2. a +b+c = *Unsymmetrical Equn. of AB in 3-D --> Take arbitrary values of c: and b x +y+z =  x +y+z = Equation of Line AB( symmetrical ) : [ (x-x1) / (x2-x1) = (y-y1) / (y2-y1) =(z-z1)/(z2-z1) ] (x- ) / =(y- ) / =(z- ) / Equn. in X-Y plane y =x +( ) Equn. in Y-Z plane z =y +( ) Equn. in Z-X plane x =z +( ) Direction Ratio of AB ,, Direction co-sines of AB (cosine of the angle w.r.t. positive  x-axis, y-axis, z-axis respectively) ,, sum of square of direction cosines (ideally should be 1) Equn. of BC in 3-D a +b+c = Unsymmetrical Equn. of BC in 3-D --> Take arbitrary values of c: and b x +y+z = - x +y+z = Equation  of line BC( symmetrical )    [ (x-x2) / (x3-x2) = (y-y2) / (y3-y2) =(z-z2)/(z3-z2) ] (x- ) / =(y- ) / =(z- ) / Equn. in X-Y plane y =x +( ) Equn. in Y-Z plane z =y +( ) Equn. in Z-X plane x =z +( ) Direction Ratio of  BC ,, Direction co-sines of BC (cosine of the angle w.r.t. positive x-axis, y-axis, z-axis respectively) ,, sum of square of direction cosines (ideally should be 1) Equn. of CA in 3-D a +b+c = Unsymmetrical Equn. of CA in 3-D --> Take arbitrary values of c: and b x +y+z = - x +y+z = Equation of line  CA( symmetrical )    [ (x-x3) / (x1-x3) = (y-y3) / (y1-y3) =(z-z3)/(z1-z3) ] (x- ) / =(y- ) / =(z- ) / Equation  in X-Y plane y =x +( ) Equation  in Y-Z plane z =y +( ) Equation  in Z-X plane x =z +( ) Direction Ratio of  CA ,, Direction co-sines of CA (cosine of the angle w.r.t. positive x-axis, y-axis, z-axis respectively) ,, sum of square of direction cosines (ideally should be 1) ** * A circle passing through the vertices of the triangle is called circumcircle. * The circle which can be inscribed within the triangle so that it touches all the 3 sides, is called the incircle. * The circle which touches the side BC and also extensions of AB and AC is called escribed circle opposite to angle A. * The circle which touches the side AC and also extensions of AB and BC is called escribed circle opposite to angle B. * The circle which touches the side AB and also extensions of BC and AC is called escribed circle opposite to angle C. * The triangle formed by joining feet of the altitudes drawn from vertices to the opposite sides is called pedal triangle. * Orthocenter/circumcenter of an acute angled triangle lies inside the triangle; that of an obtuse angle triangle lies outside the triangle. & that of right angle triangle lies at the vortex/mid-point of hypotenuse which contain right angle. *Image of the orthocentre (H) with respect to any side of triangle lies on the circumcircle. Here HK=KT where T is the image of H with respect to BC & it lies on circumcircle. * Minimum value of X is 3 i.e when a=b=c; Area= [s(s-a)(s-b)(s-c)]1/2  where s=(a+b+c)/2 or Area=absinC / 2 = bcsinA / 2 =acsinB / 2; sinA = 2*area/bc; sinB=2*area/ac ; sinC=2*area/ab; cosA=(b2 + c2- a2 ) / 2bc ; cosB=(a2 + c2- a2 ) / 2ac ; cosC=(b2 + a2- c2 ) / 2ba sin(A/2) =[ (s-c)(s-b)/bc ]1/2 cos(A/2) =[ s(s-a)/bc ]1/2 Radius of circum-circle= a/2sinA =b/2sinB=c/2sinC = abc/(4*area) Radius of incircle= area / sRadius of escribed circle ra (opposite to angle A)= area/(s-a)Radius of escribed circle rb (opposite to angle B)= area/(s-b) Radius of escribed circle rc(opposite to angle C)= area/(s-c) Circumradius of Pedal triangle= (radius of circumcircle)/2; Length of Median AD= (1/2)*[(2b2 + 2c2- a2]1/2 If H is ortho center,AH=2RcosA, HK=2RcosBcosC; BH=2RcosB;HL=2RcosAcosC,CH=2RcosC;HM=2RcosAcosB; Equn. of Plane passing through A,B,C is of the form Ax+By+Cz=D where A=y2z3-z2y3+y1z2-y2z1+z1y3-z3y1, B=x1z3-x3z1+x2z1-z2x1+x3z2-z3x2, C=x2y3-x3y2+x1y2-y1x2+y1x3-x1y3 & D=x1(y2z3-z2y3)+x2(z1y3-y1z3)+x3(y1z2-z1y2). * BK/CK=(a*a+c*c-b*b) / (a*a+b*b-c*c); *AL / CL =(b*b+c*c-a*a)/(b*b+a*a-c*c); *AM/BM=(c*c+b*b-a*a)/(c*c+a*a-b*b); *acos(B-C)+bcos(C-A)+ccos(A-B)= abc/R2  ; where cos(B-C)=cosBcosC+sinBsinC; * If a rectangle with altitude x is inscribed in triangle ABC with base b and altitude h, the perimeter P of rectangle=2[b(h-x)/x  + x] and area A = b(h-x)*x/h ; for Area A to be extremum, dA/dx=0 which implies b-2bx/h=0 or x=h/2; d2A/dx2 = -2b/h = less than zero which means that extremum is a maxima. Since  dP/dx= 0 yields b=h and does not include x, study the variation of P with variation of x. *Equation of a straight line in 2-D say x-y plane is represented as a linear equation of 2 variables such as Ax+By+C=0 where A,B,C are constants. But in 3 dimensions, the equation is 2 linear equations of 3 variables having constants  say (a3,b3,c3,d3) & (a4,b4,c4,d4). Because one equation represents a plane and a st. line is an intersection between 2 planes. Unsymmetrical Equn. of BC in 3-D (a2x+b2y+c2z+d2=0) &  (a3x+b3y+c3z+d3=0) =>a'x+b'y+c'z+d'=0 where a'=(a2-a3), b'=(b2-b3),c'=(c2-c3) & d'=(d2-d3). Putting the value of co-ordinates -> a'x2+b'y2+c'z2+d' =0 .....(1)  and  a'x3+b'y3+c'z3+d' =0 .....(2) . (1)-(2) =a'dx+b'dy+c'dz=0. Take arbitrary values of b',c'. Then a'=-(b'dy+c'dz)/dx ; If general equation of the plane on which the triangle rests is given by ax+by+cz=d , then a3=a'+a, and a4=a, b3=b'+b and b4=b and c3=c'+c and c4=c; Hence out of the 2 equations of the side of a triangle, 1 equation represents the equation of the plane on which the triangle lies and is common to all the 3 sides. And the second equation represents the unique plane of each side.* throughout the calculations, PI is nowhere used except converting degree to radian since javascript math function of sin, cos are based on radians rather than degree. Suppose, the perimeter of the triangle(p) is converted to an arc with radius x, arc length y and the task is to find out the value of x,y for which area A under the arc shall be maximum. Given p=2x+y or y=p-2x; area A=xy/2=x(p-2x)/2 . Then dA/dx=0 which means x=p/4, y=p/2. d2A/dx2 = -Ve which means above is a case of maximum area. 