ClassRoom 41 40 A Ramachandran At Right Angles At Right Angles | |
Vol. 5,No.1,March 2016 Vol. 5,No.1,March 2016 trilinear coordinates, homogeneouscoordinates, trigonometricratio Keywords: Triangle centre, incentre, centroid, orthocentre, circumcentre, PD and D Coordinates Homogeneous and Centres Triangle h nlso h triangle. of the functions of trigonometric angles and/or the lengths side the involving ratios rage rrte,terto fteedsacs hs aisfr the If form centre. ratios triangle These the distances. of these coordinates” the “trilinear of of ratios sides the three rather, in the or characterised from triangle; is distances centre perpendicular triangle its a of Here terms [1]. 1835 Germanphysicist-mathematician in by the Plücker Julius was suggested approach This coordinates Trilinear coordinates. trilinear partIof of these: In first term the later. consider of this we article, significance the the coordinate explain known ashomogeneous we but both systems, are They four these characterising points. and significant study describing shall of We ways the respectively). alternative of medians, two angles the the and of altitudes bisectors the the triangle, sides, the the of of concurrence bisectors of perpendicular points the as (introduced centroid , P PE h on nqeto,te h epniua distances perpendicular the then question, in point the , PF uring icmete h nete h rhcnr n the and orthocentre the incentre, centres—the the triangle circumcentre, classical four encounters student a otesides the to orei ulda emtya ihsho level, school high at geometry Euclidean in course a atI-TiierCoordinates Trilinear - I Part BC , CA ,
AB epcieyaeepesdas expressed are respectively
△ ABC 1 stetriangle the is
epniuast h ie,then sides, the to perpendiculars qa nae.If area. in equal ahohra h ie fteage opposite angles the to of ratios sines same the the as bear other triangle each a of sides the that Centroid. lentvl,tecodntscudb ie as given be could coordinates the Alternatively, the of coordinates centroid. trilinear the form ratios these csc ca 2k GE simply are 1 coordinates trilinear its sides, the Incentre. hsyields: This iue2.I sawl-nw euto Euclidean of triangles result that well-known geometry a is It 2). Figure : / : B 1 hence c, · ab B B b : : iue2 etodo nabtaytriangle arbitrary an of Centroid 2. Figure / seFgr 1). Figure (see 1 F F csc mliligtruhby through (multiplying 2 = ic h netei qiitn from equidistant is incentre the Since h atrlto rssfo thefact from arises relation last The C. iue1 neteo triangle; a of Incentre 1. Figure Let I A A GD GD GF seudsatfo h sides the from equidistant is G GD, · = : etecnri of centroid the be D c I GE / 2k 2 D G GE / = : a GAB, GF , E k and GE say. , E = = GF GBC 1 / abc 2 GD a are k / : and ,o scsc as or ), △ b, 1 · / ABC a GF b / GCA 2 : = 1 = (see / C C c; are A bc : : A A n sec and rhcnr r 1 are orthocentre w ftetrequantities three the of Two itntfo 0 are from assume distinct we which values limiting approach i h limit, the (in Similarly, so A, AD ofan case triangle the acute-angled consider first We orthocentre. Orthocentre. values. cosecant the same as the other bear each sides to the ratios of reciprocals the So them. tflosta o ih-nldtriangle right-angled a for that follows It cos sec A Hence: e ssewa apn soeo h nls(say angles the of 90 one approaches as A) happens what see us Let eo n ti utmr owieterto as ratios the write to customary is it and zero, rlna oriae fteotoeteaesec are orthocentre the of coordinates trilinear Since orthocentre. the , iue3 rhcnr fa ct-nldtriangle acute-angled an of Orthocentre 3. Figure B B B , A = → , F BE / C : HCD cos 90 sec , 90 B otesides the to CF EHC ◦ = HE B A ◦ C. h rlna oriae fthe of coordinates trilinear the , D cos , HD HD H HE and r epniuasfo h vertices the from perpendiculars are HC Vol. 5,No.1,March 2016 Vol. 5,No.1,March 2016 A / = o etr u teto othe to attention our turn we Now HF , ◦ = : H A / = HF n 90 and 90 HE Similarly, HD. : n sec and A, , → = ◦ E 0 ◦ BC HC HE / h te w nlsalso two angles other The . , − HD : ABC E : sec ;so 0; F , 0. HF / ◦ eget we B, CA onie e iue4). Figure see coincide; oethat Note . B HD HC = / seFgr ) Here, 3). Figure (see , HE = HD sec respectively; AB, cos = 1 A hrfr the Therefore C. → , : A HE = sec HCE sec 0 / and 0 cos | | DHC HC :
, C HE A At Right Angles At Right Angles 0 HF . . As C. / / HE. ABC = HF r now are HD = 90 → H = with A ◦ C − is B : 0 , 40 41
ClassRoom 41 40 A Ramachandran At Right Angles At Right Angles | |
Vol.Vol. 5,No.1,March 5,No.1,March 2016 2016 trilinear coordinates, homogeneouscoordinates, trigonometricratio Keywords: Triangle centre, incentre, centroid, orthocentre, circumcentre, and PD D Coordinates Homogeneous and Centres Triangle h nlso h triangle. of the functions of trigonometric angles and/or the lengths side the involving ratios rage rrte,terto fteedsacs hs aisfr the If form centre. ratios triangle These the distances. of these coordinates” the “trilinear of of ratios sides the three rather, in the or characterised from triangle; is distances centre perpendicular triangle its a of Here terms [1]. 1835 Germanphysicist-mathematician in by the Plücker Julius was suggested approach This coordinates Trilinear coordinates. trilinear partIof of these: In first term the later. consider of this we article, significance the the coordinate explain known ashomogeneous we but both systems, are They four these characterising points. and significant study describing shall of We ways the respectively). alternative of medians, two angles the the and of altitudes bisectors the the triangle, sides, the the of of concurrence bisectors of perpendicular points the as (introduced centroid , P PE h on nqeto,te h epniua distances perpendicular the then question, in point the , PF uring icmete h nete h rhcnr n the and orthocentre the incentre, centres—the the triangle circumcentre, classical four encounters student a otesides the to orei ulda emtya ihsho level, school high at geometry Euclidean in course a atI-TiierCoordinates Trilinear - I Part BC , CA , AB epcieyaeepesdas expressed are respectively △ ABC 1 stetriangle the is
epniuast h ie,then sides, the to perpendiculars qa nae.If area. in equal ahohra h ie fteage opposite angles the to of ratios sines same the the as bear other triangle each a of sides the that Centroid. lentvl,tecodntscudb ie as given be could coordinates the Alternatively, the of coordinates centroid. trilinear the form ratios these csc ca 2k GE simply are 1 coordinates trilinear its sides, the Incentre. hsyields: This iue2.I sawl-nw euto Euclidean of triangles result that well-known geometry a is It 2). Figure : / : B 1 hence c, · ab B B b : : iue2 etodo nabtaytriangle arbitrary an of Centroid 2. Figure / seFgr 1). Figure (see 1 F F csc mliligtruhby through (multiplying 2 = ic h netei qiitn from equidistant is incentre the Since h atrlto rssfo thefact from arises relation last The C. iue1 neteo triangle; a of Incentre 1. Figure Let I A A GD GD GF seudsatfo h sides the from equidistant is G GD, · = : etecnri of centroid the be D c I GE / 2k 2 D G GE / = : a GAB, GF , E k and GE say. , E = = GF GBC 1 / abc 2 GD a are k / : and ,o scsc as or ), △ b, 1 · / ABC a GF b / GCA 2 : = 1 = (see / C C c; are A bc : : A A n sec and rhcnr r 1 are orthocentre w ftetrequantities three the of Two itntfo 0 are from assume distinct we which values limiting approach i h limit, the (in AD ofan case triangle the acute-angled consider first We orthocentre. Orthocentre. values. cosecant the same as the other bear each sides to the ratios of reciprocals the So them. Similarly, so A, tflosta o ih-nldtriangle right-angled a for that follows It cos sec A Hence: e ssewa apn soeo h nls(say angles the of 90 one approaches as A) happens what see us Let eo n ti utmr owieterto as ratios the write to customary is it and zero, rlna oriae fteotoeteaesec are orthocentre the of coordinates trilinear Since orthocentre. the , iue3 rhcnr fa ct-nldtriangle acute-angled an of Orthocentre 3. Figure B B B , A = → , F BE / C : HCD cos 90 sec , 90 B otesides the to CF EHC ◦ = HE B A ◦ C. h rlna oriae fthe of coordinates trilinear the , D cos , HD HD H HE and r epniuasfo h vertices the from perpendiculars are HC Vol. 5,No.1,March 2016 Vol. 5,No.1,March 2016 A / = o etr u teto othe to attention our turn we Now HF , ◦ = : H A / = HF n 90 and 90 HE Similarly, HD. : n sec and A, , → = ◦ E 0 ◦ BC HC HE / h te w nlsalso two angles other The . , − HD : ABC E : sec ;so 0; F , 0. HF / ◦ eget we B, CA onie e iue4). Figure see coincide; oethat Note . B HD HC = / seFgr ) Here, 3). Figure (see , HE = HD sec respectively; AB, cos = 1 A hrfr the Therefore C. → , : A HE = sec HCE sec 0 / and 0 cos | | DHC HC :
, C HE A At Right Angles At Right Angles 0 HF . . As C. / / HE. ABC = HF r now are HD = 90 → H = with A ◦ C − is B : 0 , 40 41 A, H, E, F A A A
F E F E E F B D C O B D C B D, O C O Figure 4. Orthocentre of a right-angled triangle
Figure 7. Circumcentre of a right-angled triangle Figure 8. Circumcentre of an obtuse-angled triangle H B D C
F obtained earlier; only, cos A has now assumed a Note from the editor: What is homogeneous zero value. about this system? The significance of the term In the case of an obtuse-angled triangle, the homogeneous may not be immediately apparent. In Figure 6. Circumcentre of an acute-angled triangle circumcentre O lies outside the triangle (see ‘ordinary’ coordinate geometry, the equation of a E + + = , , Figure 8, where A is obtuse). line has the form ax by c 0, where a b c A are constants. Note that this equation is not and we find that the inherent symmetry ofthe Here we have BOD = 180◦ A, so OD = − homogeneous: two terms have degree 1, while one formula has been restored. R cos A. The relations OE = R cos B and OF = − term has degree 0. Similarly, the question of a B D C R cos C remain unchanged. So we get OD : OE : circle has the form x2 + y2 + 2gx + 2fy + c = 0; Circumcentre. Next, we consider the OF = cos A : cos B : cos C. If we adopt the this too is not homogeneous. In some settings, it Figure 5. Orthocentre of an obtuse-angled triangle − circumcentre. We first look at an acute-angled same sign convention as earlier, then OD < 0 turns out to be advantageous to have equations triangle ABC (see Figure 6). since O and A lie on opposite sides of BC, while which are homogeneous, in which all the terms OE > 0 and OF > 0, since O and B lie on the have the same degree. The trilinear coordinates Next, consider the case of an obtuse-angled In the figure, O is the circumcentre of ABC, and △ same side of CA, and O and C lie on the same side system described above has this feature, and so triangle with say A as the obtuse angle (see OD is perpendicular to BC. We have: BOC = of AB. With this understanding, the symmetry of does the barycentric coordinates system to be Figure 5); we will find that a negative sign appears 2 A, hence BOD = A and cos A = the formula gets restored and we have: OD : OE : discussed in part II. Here, the equation of a line in the trilinear relationship. OD/OB = OD/R, where R is the circumradius. OF = cos A : cos B : cos C. So the trilinear has the form lx + my + nz = 0, where l, m, n are Thus OD = R cos A. It follows that the distances We have: EHC = CAF = 180◦ A. Then coordinates of the circumcentre are cos A : cos B : constants; note that this equation is homogeneous. − from O to the sides of the triangle are proportional HC/HE = sec(180◦ A)= sec A. Now, cos C. In recent times it has been found that − − to the cosines of the angles opposite them. Hence HC/HD = sec B (since DHC = B), so In Part II of the article, we shall describe another homogeneous coordinates are particularly HD/HE = sec A/sec B. However, HE/HF = the trilinear coordinates of the circumcentre are − : : such coordinate system—barycentric coordinates. convenient to use in computer graphics. sec B/sec C, while HF/HD = sec C/sec A. So cos A cos B cos C. − we get: Just as we did last time, let us see what happens as References one angle (say A) approaches 90◦. The other two HD : HE : HF = sec A : sec B : sec C. 1. Julius Plücker. http://www-groups.dcs.st-and.ac.uk/ history/Biographies/Plucker.html − angles also approach limiting values which we 2. Wikipedia, Homogeneous Coordinates. https://en.wikipedia.org/wiki/Homogeneous_coordinates There is a way of looking at this relationship which assume are distinct from 0◦ and 90◦. restores the symmetry of signs. Note that if A > The situation is depicted in Figure 7; D and O 90◦ (as in Figure 5), then H and A lie on the same now coincide, and cos A = OD = 0. Also side of BC, while H and B lie on opposite sides of CA, and similarly, H and C lie on opposite sides of OE AB/2 AB/BC cos B AB. Recalling the sign convention for distances = = = , used in coordinate geometry, we see that it makes OF AC/2 AC/BC cos C A. RAMACHANDRAN has had a longstanding interest in the teaching of mathematics and science. He studied sense to regard HD as positive, and HE and HF as physical science and mathematics at the undergraduate level, and shifted to life science at the postgraduate level. He so OD : OE : OF = 0 : cos B : cos C. Hence the taught science, mathematics and geography to middle school students at Rishi Valley School for two decades. His negative. Under this perspective, we have: trilinear coordinates of the circumcentre are 0 : other interests include the English language and Indian music. He may be contacted at [email protected]. HD : HE : HF = sec A : sec B : sec C, cos B : cos C. This is consistent with the formula
4342 At Right Angles | Vol. 5, No. 1, March 2016 Vol. 5, No. 1, March 2016 | At Right Angles 4342 A, H, E, F A A A
F E F E E F B D C O B D C B D, O C O Figure 4. Orthocentre of a right-angled triangle
Figure 7. Circumcentre of a right-angled triangle Figure 8. Circumcentre of an obtuse-angled triangle H B D C
F obtained earlier; only, cos A has now assumed a Note from the editor: What is homogeneous zero value. about this system? The significance of the term In the case of an obtuse-angled triangle, the homogeneous may not be immediately apparent. In Figure 6. Circumcentre of an acute-angled triangle circumcentre O lies outside the triangle (see ‘ordinary’ coordinate geometry, the equation of a E + + = , , Figure 8, where A is obtuse). line has the form ax by c 0, where a b c A are constants. Note that this equation is not and we find that the inherent symmetry ofthe Here we have BOD = 180◦ A, so OD = − homogeneous: two terms have degree 1, while one formula has been restored. R cos A. The relations OE = R cos B and OF = − term has degree 0. Similarly, the question of a B D C R cos C remain unchanged. So we get OD : OE : circle has the form x2 + y2 + 2gx + 2fy + c = 0; Circumcentre. Next, we consider the OF = cos A : cos B : cos C. If we adopt the this too is not homogeneous. In some settings, it Figure 5. Orthocentre of an obtuse-angled triangle − circumcentre. We first look at an acute-angled same sign convention as earlier, then OD < 0 turns out to be advantageous to have equations triangle ABC (see Figure 6). since O and A lie on opposite sides of BC, while which are homogeneous, in which all the terms OE > 0 and OF > 0, since O and B lie on the have the same degree. The trilinear coordinates Next, consider the case of an obtuse-angled In the figure, O is the circumcentre of ABC, and △ same side of CA, and O and C lie on the same side system described above has this feature, and so triangle with say A as the obtuse angle (see OD is perpendicular to BC. We have: BOC = of AB. With this understanding, the symmetry of does the barycentric coordinates system to be Figure 5); we will find that a negative sign appears 2 A, hence BOD = A and cos A = the formula gets restored and we have: OD : OE : discussed in part II. Here, the equation of a line in the trilinear relationship. OD/OB = OD/R, where R is the circumradius. OF = cos A : cos B : cos C. So the trilinear has the form lx + my + nz = 0, where l, m, n are Thus OD = R cos A. It follows that the distances We have: EHC = CAF = 180◦ A. Then coordinates of the circumcentre are cos A : cos B : constants; note that this equation is homogeneous. − from O to the sides of the triangle are proportional HC/HE = sec(180◦ A)= sec A. Now, cos C. In recent times it has been found that − − to the cosines of the angles opposite them. Hence HC/HD = sec B (since DHC = B), so In Part II of the article, we shall describe another homogeneous coordinates are particularly HD/HE = sec A/sec B. However, HE/HF = the trilinear coordinates of the circumcentre are − : : such coordinate system—barycentric coordinates. convenient to use in computer graphics. sec B/sec C, while HF/HD = sec C/sec A. So cos A cos B cos C. − we get: Just as we did last time, let us see what happens as References one angle (say A) approaches 90◦. The other two HD : HE : HF = sec A : sec B : sec C. 1. Julius Plücker. http://www-groups.dcs.st-and.ac.uk/ history/Biographies/Plucker.html − angles also approach limiting values which we 2. Wikipedia, Homogeneous Coordinates. https://en.wikipedia.org/wiki/Homogeneous_coordinates There is a way of looking at this relationship which assume are distinct from 0◦ and 90◦. restores the symmetry of signs. Note that if A > The situation is depicted in Figure 7; D and O 90◦ (as in Figure 5), then H and A lie on the same now coincide, and cos A = OD = 0. Also side of BC, while H and B lie on opposite sides of CA, and similarly, H and C lie on opposite sides of OE AB/2 AB/BC cos B AB. Recalling the sign convention for distances = = = , used in coordinate geometry, we see that it makes OF AC/2 AC/BC cos C A. RAMACHANDRAN has had a longstanding interest in the teaching of mathematics and science. He studied sense to regard HD as positive, and HE and HF as physical science and mathematics at the undergraduate level, and shifted to life science at the postgraduate level. He so OD : OE : OF = 0 : cos B : cos C. Hence the taught science, mathematics and geography to middle school students at Rishi Valley School for two decades. His negative. Under this perspective, we have: trilinear coordinates of the circumcentre are 0 : other interests include the English language and Indian music. He may be contacted at [email protected]. HD : HE : HF = sec A : sec B : sec C, cos B : cos C. This is consistent with the formula
4342 At Right Angles | Vol. 5, No. 1, March 2016 Vol. 5, No. 1, March 2016 | At Right Angles 4342