Insight with Geometry xpressions INSIGHT WITH GOMTRY XPRSSIONS... 1 INTROUTION... WRM UP... 4 xmple 1: sequence of ltitudes... 5 xmple : ngles nd ircles... 10 xmple 3: Tringultion... 13 xmple 4: Rectngle ircumscriing n quilterl Tringle... 17 xmple 5: re of Hexgon ounded y Trigle side trisectors... 18 N INVSTIGTION OF INIRLS, IRUMIRLS N RLT MTTRS... 1 xmple 6: ircumcircle Rdius... xmple 7: Incircle Rdius... 9 xmple 8: Incircle enter in rycentric oordintes... 31 xmple 9: How does the point of contct with the incircle split line... 34 xmple 10: xcircles... 35 1
INSIGHT WITH GOMTRY XPRSSIONS Introduction Geometry xpressions utomticlly genertes lgeric expressions from geometric figures. For exmple in the digrm elow, the user hs specified tht the tringle is right nd hs short sides length nd. The system hs clculted n expression for the length of the ltitude: + n system which does geometry e used in teching geometry? I d suggest the nswer is yes. nd this rticle is my ttempt to justify tht nswer y exmple. In 1981, I ws supplementing my income s reserch ssistnt y tutoring mthemtics. One of my students ws studying for his ity nd Guild exm in electronics. The exmintion s contents hd not een updted in decde, nd prt of the exm ws to perform multipliction nd power clcultions using log tles. Mthemticins of my ge or older will rememer the peculir rithmetic used with log tles where numers less thn 1 were involved. y 198 the dvent of inexpensive pocket clcultors hd completely eliminted the need to use log tles. However the exm still contined these
prolems, nd my student hd to lern how to do them, except insted of looking up log nd ntilog tles, he used the log function on his clcultor. Technology, I contend, should e emrced rther thn ignored. 3
INSIGHT WITH GOMTRY XPRSSIONS Wrm Up The min section of this rticle is n investigtion of specific geometric topic (Incircles nd ircumcircles) using Geometry xpressions. s wrm up, we ll exmine hndful of simple exmples in which we hope to show how Geometry xpressions cn e s prt of process of cretive mthemtics. 4
xmple 1: sequence of ltitudes We re going to look t the figure we showed in the introduction: s this is xmple 1, I ll show you in some detil how to crete this digrm: We will use three toolrs, the rw toolr to crete the geometry, the onstrin toolr to specify lengths nd ngles, nd the lculte toolr to mesure the length of the ltitude. First however, we sketch the figure using the line segment tool: Next, dd the right ngle constrints nd the length constrints for lines nd : 5
INSIGHT WITH GOMTRY XPRSSIONS Finlly, clculte the length of, y selecting it then clicking the lculte Symolic Length utton: 6
n you prove tht the expression for the ltitude is correct? (Think out the reltionship etween ltitude nd re for tringle). Now let s look t the length : + + Wht is the rtio /? Wht does this tell you out the reltionship etween the tringle nd the tringle? n you estlish this reltionship in different wy (think ngles)? Hence, cn you prove the formul for? Wht is the length? Wht is the rtio /? Wht is the reltionship etween tringles nd? Wht is the rtio of the hypotenuse of the two tringles? Now let s crete the incircles for nd. (To crete them, sketch the circles, then pply tngent constrints etween the circles nd the pproprite sides of the tringles) 7
INSIGHT WITH GOMTRY XPRSSIONS H F G Wht is the rtio of the rdii of these incircles? H F G z 4 + + + + + z 5 - - + -- + - + z 4 z 5 We see the rtio is /. Is this surprise? Let s go ck to our originl drwing nd crete nother ltitude: 8
- - - Wht is the rtio /? n you prove the formul for? n you predict the length of FG in the drwing elow? G - - - F 9
INSIGHT WITH GOMTRY XPRSSIONS xmple : ngles nd ircles If chord sutends n ngle of θ t the center of circle wht does it sutend t the circumference? θ θ n you prove this result. (Hint: strt with the digrm elow nd fill in the ngles). φ θ 10
Here s sequence of digrms, does this constitute proof? π-θ-φ π-θ-φ φ θ π - φ φ θ π - φ -π+θ+φ π - φ π-θ-φ φ θ π - φ 11
INSIGHT WITH GOMTRY XPRSSIONS n you follow similr pproch nd prove these results: θ π- θ θ - φ F φ θ 1
xmple 3: Tringultion Geometry xpressions hs its own lger system customized to mnipulting the sort of mthemtics tht rises in geometry prolems. However, to do further nlysis of your geometry expressions you should copy them into more fully fetured lger system. Geometry xpressions exports expressions in the MthML formt, which is ccepted s input y wide vriety of mthemtics disply nd computtion pplictions. In this exercise, we will work on n exmple which involves copying into n lger system to complete the nlysis. ssume we re writing computer progrm which performs tringultion: given the length of seline nd ngles mesured off the seline to the pex of the tringle, the progrm is to give the coordintes of the pex. The pproprite expressions cn e derived from Geometry xpressions: - +c z 0 c z 1 + - 4 + - 4 - c 4 c c 4 c 4 +c > c In the computer progrm, we wish to know wht the error in the derived quntities is reltive to error in the mesured quntities nd. 13
INSIGHT WITH GOMTRY XPRSSIONS rror in z 0 due to n error δ in is pproximtely: δz 0 δ dz0 d nd similrly for error in z 1 dz The quntity 0 d cn e thought of s the error mgnifier. If you select the expression z 0 you cn copy nd pste into n lger system: Mple, or Mthemtic or ny other system which is prepred to ccept mthml. We ll use the sio lsspd Mnger. Once in there, we ll differentite with respect to to get n expression for the error mgnifiction: The error mgnifier in z 0 is simply the rtio of /c. In z 1 the mgnifier is more complicted. Fctoring the term under the squre root gives us clerer picture: We see tht the denomintor of the error term goes to zero when =+c, or when =+c, or when c=+. n you interpret these conditions geometriclly? 14
n you interpret the complete error term geometriclly? Hint: compre the denomintor with the re of the tringle. ompre the numertor with the distnce in the digrm. n you construct distnce on the digrm whose length is more closely relted to the numertor? One question we might sk is this: wht is the optiml geometry for tringulting? Let s simplify the question y ssuming the tringle is isosceles nd ==x. We ll lso ssume the se length is 1. We cn grph these functions nd oserve tht for lrge vlues of x the first domintes, for smll vlues of x the second domintes. 15
INSIGHT WITH GOMTRY XPRSSIONS The optiml vlue of x is when the two re equl: Wht ngles does this tringle hve? 16
xmple 4: Rectngle ircumscriing n quilterl Tringle Iin pge 19-1 of Mthemticl Gems, y Ross Honserger (nd vrious other plces), we hve the following theorem: Inscrie n equilterl tringle in squre such tht one corner of the tringle is corner of the squre nd the other two corners lie on the opposite sides of the squre. This forms 3 right tringles. The theorem sttes: The re of the lrger right tringle is the sum of the res of the smller two. You cn mesure res in Geometry xpressions y selecting connected set of segments nd constructing polygon. You cn then crete n re mesure for tht polygon. The digrm elow shows the res of the tringles in the ove theorem - 3 8-3 4 - - 4 3 8-3 4 + - 4 F n you prove the theorem from the digrm? n you prove tht the digrm is correct? 17
INSIGHT WITH GOMTRY XPRSSIONS xmple 5: trisectors re of Hexgon ounded y Trigle side 3 3 ++c +-c -+c -++c 40 H F K 3 I J O N L M G 3 c 3 c c 3 The shded hexgon is formed y intersecting the lines joining the vertices of the tringle with the trisectors of the opposite sides. How does the re relte to the re of the tringle? n you prove this? One pproch uses this expression for the loction of the intersection of nd where is proportion t long nd is proportion t long. Use vlues 1/3 nd /3 for t to get points of the hexgon 18
x 1,y 1 t G x 0,y 0 F 1-t -x 0 +t x 0 -x 1 +t x 1 -t x -+t -y 0 +t y 0 -y 1 +t y 1 -t y, -+t x,y n you prove the re of the hexgon using this informtion? n you prove the coordintess? 19
INSIGHT WITH GOMTRY XPRSSIONS 0 How out the re of the other hexgon? N L I M K J F G O H 13 ++c +-c -+c -++c 196 3 3 3 c 3 c 3 c 3
n Investigtion of Incircles, ircumcircles nd relted mtters We ll follow with set of exmples ll relted to the theme of incircles, circumcircles, excircles nd tringle res. 1
INSIGHT WITH GOMTRY XPRSSIONS xmple 6: ircumcircle Rdius Mesure the re of tringle sides length,,c, nd mesure the rdius of the circumcircle. Tringle re ircumcircle rdius c ++c +-c -+c -++c ++c +-c -+c -++c 4 c Wht is their reltionship? If tringle is defined in terms of two sides nd the included ngle, wht is its re? Hence, derive formul for the rdius of the circumcircle which involves n ngle:
You cn see this is Geometry xpressions: φ + - cos(φ) sin(φ) + - cos(φ) 3
INSIGHT WITH GOMTRY XPRSSIONS n you prove tht this expression is true independent of Geometry xpressions? Hint, in the digrm elow, get n expression for r θ θ Let R e the rdius of the circumcircle, let,, e the ngles of tringle whose opposite sides hve length,,c. t this point, you should hve proved: R = c sin( ) 1 re( ) = sin( ) = c 4R Hence: c R = 4re( ) 4
Now to prove the originl formul for R in terms of,,c, we need only prove (or ccept without proof) the formul for the re of the tringle generted y Geometry xpressions. We don t like to ccept nything without proof do we? - +c c c - + +c c Show tht with the ove vlues for nd, = + = c opy the expression for into your lger system. Now crete n expression for the squre of the ltitude. Multiply y the squre of c, nd divide y 4 to crete n expression for the squre of the re: > c^*(^-(1/*(^-^+c^)/c)^)/4; 1 4 c - ( - + c ) 4 c 5
INSIGHT WITH GOMTRY XPRSSIONS You cn then do some lgeric mnipultion to get this into nicer form: > expnd(%); 1 8 c - 1 16 4 + 1 8-1 16 4 + 1 8 c - 1 16 c 4 > simplify(%); 1 8 c - 1 16 4 + 1 8-1 16 4 + 1 8 c - 1 16 c 4 > fctor(%); - 1 16 ( + + c) ( + - c) (-c + - ) ( - + c) 6
s finl exercise, let s look t the center of the circumcircle in terms of the coordintes of the tringle vertices (rycentric coordintes) y 1 -y x 0 +x 1 x 0 -x 1 + y 0 +y 1 y 0 -y 1 x 1 +x -x 1 +x y 1 +y -y 1 +y + y 0 -y 1 + - x 1 -x, -x 1 y 0 +x y 0 +x 0 y 1 -x y 1 -x 0 y +x 1 y x 0 +x 1 x 1,y 1 x 1 y 0 -x y 0 -x 0 y 1 +x y 1 +x 0 y -x 1 y x 0,y 0 x,y The expression is quite complicted, ut reks down into constituent prts. o you see the re emedded in the formul? 7
INSIGHT WITH GOMTRY XPRSSIONS n you write this simpler in vector terms? -u v0 1 -v 0 v 1 +v1 u +v0 0 -u u1 0 +u 0 u 1 -u1 v 0 +u0 v 1 -u 1 v 0 +u 0 v 1, -u 1 v 0 +u 0 v 1 u 0 v 0 (0,0) u 1 v 0 u 0 v - 1 u 1 v 1 8
xmple 7: Incircle Rdius Here is the formul for the incircle rdius (long with the fmilir formul for the re of the tringle). z 1 ++c +-c -+c -++c 4 +-c -+c -++c ++c c n you express the incircle rdius in terms of the re? Now cn you prove it independently of Geometry xpressions? 9
INSIGHT WITH GOMTRY XPRSSIONS Hint: consider the res of the shded tringle: r c 30
xmple 8: Incircle enter in rycentric oordintes x 1,y 1 x 0,y 0 x,y x 1 x 0 -x + -y0 +y +x -x 0 +x 1 + y0 -y 1 +x0 -x 1 +x + y1 -y x 0 -x + -y0 +y + -x0 +x 1 + y0 -y 1 + -x1 +x + y1 -y y 1 x 0 -x + -y0 +y +y -x 0 +x 1 + y0 -y 1 +y0 -x 1 +x + y1 -y, x 0 -x + -y0 +y + -x0 +x 1 + y0 -y 1 + -x1 +x + y1 -y If we let =, = nd c= nd let, nd e the position vectors of the points,, then the incircle center is: + + c + + c rete point with rycentric coordintes (r,s,1-r-s) nd exmine the res of the 3 tringles defined y the point nd the three sides of the tringle. (you should lock the prmeter vlues r nd s to some vlues such tht r+s<1, then the point will lie inside the tringle): 31
INSIGHT WITH GOMTRY XPRSSIONS x 1,y 1 z 1 x 1 y 0 -x y 0 -x 0 y 1 +x y 1 +x 0 y -x 1 y r x 0 +s x 1 +x (1-r-s),r y 0 +s y 1 +y (1-r-s) x 0,y 0 x,y z 0 s x 1 y 0 -x y 0 -x 0 y 1 +x y 1 +y x 0 -x 1 hs rycentric coordintes (r,s,1-r-s) Note tht the rtio of the res / is the rycentric coordinte s Wht is the rtio of this re to the re of the originl tringle? n you use this reltionship to prove the formul for the rycentric coordintes of the incenter? 3
n you use this reltionship to express the rycentric coordintes of the circumcenter in terms of the lengths of the sides of the tringle? -c - - +c 4 ++c +-c -+c -++c c 33
INSIGHT WITH GOMTRY XPRSSIONS xmple 9: split line How does the point of contct with the incircle +-c H F -+c G c n you deduce the lengths H, H, G, G? 34
xmple 10: xcircles The three excircles of tringle re tngent to the three sides ut exterior to the circle: F G I c ++c +-c -++c -+c H Wht is the rtio etween this rdius nd the re of the tringle? n you prove the result using res of tringles F, F,F? Wht re the rdii of the other two excircles? Wht is the product of the rdii of the three excircles nd the incircle? 35
INSIGHT WITH GOMTRY XPRSSIONS We exmine the tringle joining the centers of the excircles: c (-+c) (-++c) F G I c ++c +-c -++c -+c H n you prove tht lies on F (think in terms of symmetry)? n you prove the length? 36