Circles and Tangents with Geometry Expressions

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Britton Logan Tyler
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1 icles nd Tngents with eomety xpessions IRLS N TNNTS WITH OMTRY XPRSSIONS... INTROUTION... 2 icle common tngents... 3 xmple : Loction of intesection of common tngents... 4 xmple 2: yclic Tpezium defined by common tngents of 2 cicles... 5 xmple 3: Tingle fomed by the intesection of the inteio common tngents of two cicles... 7 xmple 4: Locus of centes of common tngents to two cicles... 8 xmple 5: Length of the common tngent to two tngentil cicles... 9 xmple 6: Tngents to the Rdicl xis of Pi of icles... 0 belos... xmple 7: Vious icles in n quiltel Tingle... 2 xmple 8: Two cicles inside cicle twice the dius, then thid... 4 xmple 9: theoem old in Pppus time... 9 xmple 0: nothe mily of Tngentil icles xmple : Yet nothe mily of icles xmple 2: chimedes Twins xmple 3: Squeezing cicle between two cicles... 29

2 IRLS N TRINLS WITH OMTRY XPRSSIONS Intoduction eomety xpessions utomticlly genetes lgebic expessions fom geometic figues. o exmple in the digm below, the use hs specified tht the tingle is ight nd hs shot sides length nd b. The system hs clculted n expession fo the length of the ltitude: b b 2 +b 2 We pesent collection of woked exmples using eomety xpessions. In most cses, digm is pesented with little comment. It is hoped tht these digms e sufficiently self explntoy tht the ede will be ble to cete them himself. The gol of these exmples is to demonstte the sot of poblems which the softwe is cpble of hndling, nd to suggest venues of futhe explotion fo the ede. The exmples e clusteed by theme. 2

3 icle common tngents The following set of exmples exploes some popeties of the common tngents of pis of cicles. 3

4 IRLS N TRINLS WITH OMTRY XPRSSIONS xmple : Loction of intesection of common tngents icles nd hve dii nd s espectively. If the centes of the cicles e pt, nd is the intesection of the inteio common tngent with the line joining the two centes, wht e the lengths nd? s +s s +s How bout the exteio common tngent? s - -+s s< - s -+s s< 4

5 xmple 2: yclic Tpezium defined by common tngents of 2 cicles iven cicles dii nd s nd distnce pt, wht is the ltitude of the tpezium fomed by joining the intesections of the 4 common tngents with one of the cicles? 2 s 2 s I s H J Notice tht this is symmeticl in nd s, nd hence the tpezium in cicle hs the sme ltitude. 5

6 IRLS N TRINLS WITH OMTRY XPRSSIONS Look t the tio of the es of the tpezi in the pevious digm: z 2 2 s s-s s s-s 2 2 z 2 2 s s-s 2 +2 s s-s 2 2 I s H J z z 2 s Notice tht the tio of tpezium es is the sme s the tio of dii. 6

7 xmple 3: Tingle fomed by the intesection of the inteio common tngents of two cicles Notice tht if is the e of the tingle fomed by the centes of the cicles, then e STU is: 2st ( + s)( s + t)( + t) s s t +b+c +b-c -b+c -+b+c 2 (+s) (+t) (s+t) P I c W O K X M H t Q Y J b L N Notice tht this tio is independent of the size of,b, nd c. 7

8 IRLS N TRINLS WITH OMTRY XPRSSIONS xmple 4: Locus of centes of common tngents to two cicles W tke the locus s the dius of the left cicle vies. The midpoints of ll fou common tngents lie on the sme fouth ode cuve 4 X 4 +8 X 2 Y 2 +4 Y 4-2 X 3-2 X Y s 2 +Y s 2 +X s 2 +X s 2 =0 K s L (0,0) (,0) M I H R J We cn use Mple to solve fo the intesections with the x xis: > subs(y=0,4*x^4+8*y^2*x^2+4*y^4-2**x^3-2**y^2*x+^4-s^2*^2+(4*^2-4*s^2)*y^2+(3*^2-4*s^2)*x^2+(-6*^3+4*s^2*)*x ); 4 X 4-2 X s ( s 2 ) X 2 + ( s 2 ) X > solve(%,x); - s, + s, 2, 2 8

9 xmple 5: Length of the common tngent to two tngentil cicles succinct fomul: 2 s s 9

10 IRLS N TRINLS WITH OMTRY XPRSSIONS xmple 6: Tngents to the Rdicl xis of Pi of icles The dicl xis of pi of cicles is the line joining the points of intesection. The lengths of tngents fom given point on this xis to the two cicles e the sme b s 4 +s b s 4 +s b H s 0

11 belos set of exmples studying cicles squeezed between othe cicles.

12 IRLS N TRINLS WITH OMTRY XPRSSIONS xmple 7: Vious icles in n quiltel Tingle We look t the dii of vious cicles in n equiltel tingle:

13 3 54 I H Wht would the next length in the sequence be? 3

14 IRLS N TRINLS WITH OMTRY XPRSSIONS xmple 8: then thid Two cicles inside cicle twice the dius, H

15 nd if we keep on going: 2 3 H J 3 2 I L K 9 M N 2 5

16 IRLS N TRINLS WITH OMTRY XPRSSIONS The genel cse looks like this: 2 + x x x H 2 We cn copy this expession into Mple to genete the bove sequence: > /(/2*/+/x+2*sqt(-/2*/(^2)+/2/x/)); > subs(=,%); x x 6

17 + + 2 x 2 + > f:=x->/(/2+/x+sqt(-2+2/x)); f := x x > f(); > f(2/3); > f(/3); > f(2/); > f(/9); > f(2/27); > little nlysis of the seies cn led us to postulte the fomul 2/(n^2+2) fo the n th cicle: Let s feed the n-th tem into Mple: > f(2/((n-)^2+2)); 2 x 2 x 7

18 IRLS N TRINLS WITH OMTRY XPRSSIONS ( n ) 2 ( n ) 2 In ode to get the expession to simplify, we mke the ssumption tht n>: > ssume(n>); > simplify(f(2/((n-)^2+2))); n~ 2 We see tht this is the next tem in the seies. y induction, we hve shown tht the n th 2 cicle hs dius n 8

19 xmple 9: theoem old in Pppus time theoem which ws old in Pppus dys (3 d centuy ) eltes the dii to height of the cicles in figues like the bove: 2 3 H J 3 2 I L K

20 IRLS N TRINLS WITH OMTRY XPRSSIONS pplying the genel model, we get fomul: x 2 2 x 2-2 x gin, we cn copy this into Mple fo nlysis: > subs(x=2/(n^2+2),=,%); 2 x 2 2 x 20

21 > simplify(%); n~ n~ 2 4 n~ 2 + n~ 2 We see tht the height bove the centeline fo these cicles is the dius multiplied by 2n. 2

22 IRLS N TRINLS WITH OMTRY XPRSSIONS xmple 0: nothe mily of Tngentil icles H I K L J 3 22 N M

23 We cn follow though simil nlysis to the pevious section: 3 + x x x H 3 2 > f:=x->/(/3+/x+2/3*sqt(-6+3/x)); f := x x 3 x > f(); I 3 23

24 IRLS N TRINLS WITH OMTRY XPRSSIONS > f(3/0); > f(%); > > ssume(n>); > simplify(f(3/((n-)^2+6))); n~ 2 > simplify(%); 24

25 xmple : Yet nothe mily of icles We genelize the sitution fom couple of exmples go. We look t the fmily geneted by two cicles of dius nd b inside cicle of dius +b: b (+b) b+b 2 H b (+b) 2 + b+b 2 b (+b) J b+b 2 b (+b) b+b 2 N K L I +b M b The ptten is petty obvious this time: the dius of the nth cicle is: b( + b) 2 2 n + b + b 2 25

26 IRLS N TRINLS WITH OMTRY XPRSSIONS To pove this, we deive the fomul fo the genel cicle dius x, nd nlyze in Mple: b + x - +b +2 b x - b (+b) - x (+b) H J I x +b b Now we ty feeding in one of the cicle dii into this fomul in mple (fist mking the ssumption tht the dii e positive (long with n> fo lte use): > ssume(>0,b>0,n>); > f:=x->/(/b+/x-/(+b)+2*sqt(/(x*b)-/((+b)*b)- /((+b)*x))); f := x b x + b xb ( + b) b ( + b) x 26

27 > f((+b)*b*/(9*^2+b*+b^2)); 9 ~ 2 + b~ ~ + b~ 2 / + b~ ( ~ + b~ ) b~ ~ ~ + b~ 9 ~ 2 + b~ ~ + b~ ( ~ + b~ ) b~ 2 ~ ( ~ + b~ ) b~ > simplify(%); ( ~ + b~ ) b~ ~ 6 ~ 2 + b~ ~ + b~ 2 Let s ty the genel cse, feeding in the fomul fo the n- st dius: 9 ~ 2 + b~ ~ + b~ 2 ( ~ + b~ ) 2 b~ ~ > simplify(f((+b)*b*/((n-)^2*^2+b*+b^2))); b~ ~ ( ~ + b~ ) b~ ~ + ~ 2 n~ 2 + b~ 2 y induction, we hve poved the genel esult. 27

28 IRLS N TRINLS WITH OMTRY XPRSSIONS xmple 2: chimedes Twins The given cicles e mutully tngentil with dius, b nd +b. chimedes twins e the cicles tngentil to the common tngent of the inne cicles. We see fom the symmety of the dius expession tht they e conguent. +b b +b I b +b K b 28

29 xmple 3: Squeezing cicle between two cicles Tke cicle dius 2 centeed t (,0) nd cicle dius 4 centeed t (-,0). Now look t the locus of the cente of the cicle tngent to both. 4 t x 2-9 y 2 =0 (-,0) (,0) 2 It s n ellipse. om the dwing we cn see tht the semi mjo xis in the x diection is 3. Wht is the semi mjo xis in the y diection? 29

r (1+cos(θ)) sin(θ) C θ 2 r cos θ 2

r (1+cos(θ)) sin(θ) C θ 2 r cos θ 2 icles xmple 66: Rounding one ssume we hve cone of ngle θ, nd we ound it off with cuve of dius, how f wy fom the cone does the ound stt? nd wht is the chod length? (1+cos(θ)) sin(θ) θ 2 cos θ 2 xmple 67: