Thomas J. Osler, Mathematics Department and Tirupathi R. Chandrupatla, Mechanical Engineering. Rowan University Glassboro, NJ
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1 /9/04 SOME UNUSUAL EXPRESSIONS FOR THE INRADIUS OF A TRIANGLE Tomas J. Osle, Matematics Depatment and Tiupati R. Candupatla, Mecanical Engineeing Rowan Univesity Glassboo, NJ 0808 Osle@owan.edu Candupatla@owan.edu Abstact Seveal fomulae fo te inadius of vaious types of tiangles ae deived. Popeties of te inadius and tigonometic functions of te angles of Pytagoean and Heonian tiangles ae also pesented. Te entie pesentation is elementay and suitable fo classes in geomety, pecalculus matematics and numbe teoy.
2 /9/04 SOME UNUSUAL EXPRESSIONS FOR THE INRADIUS OF A TRIANGLE Te incicle of a tiangle is tat cicle wic just touces all tee sides of te tiangle. Figue sows te incicle fo a tiangle. It is easy to see tat te cente of te incicle (incente) is at te point wee te angle bisectos of te tiangle meet. α / c b β / β / a γ / γ / Figue Te pupose of tis sot note is to deive seveal elations fo te inadius of a tiangle. In addition we use tese to exploe some facts egading Pytagoean and Heonian tiangles. Teoem [] is found in []. We wee unable to locate Teoems, 6, 7and 8 in te liteatue. Teoem A is poved in [3], Teoems 3 and 4 ae found in [], and Teoem 5 is essentially poved in [4]. We included tem and tei poofs to make te pesentation complete. Te entie pesentation is elementay and suitable fo classes in geomety, pecalculus matematics and numbe teoy. Teoem : Let te sides of a tiangle be a, b and c. Ten te inadius is given by
3 3 () A =. a + b + c Poof: Refe to Figue. Te aea A of te tiangle is te sum of te aeas of te tee inteio tiangles made by te angle bisectos. Tis sum is a b c a + b + c A= + + = Solving tis last elation fo we get (). Teoem : Let te sides of a tiangle be a, b and c, and let te opposite angles be αβ, and γ. Ten te inadius is given by te following tee expessions. b+ c a α () = tan a+ c b β (3) = tan a+ b c γ (4) = tan Poof: Te altitude of te tiangle is () becomes acsin β =. a + b + c Multiplying and dividing by ( a+ c b) we get = csin β, so twice te aea is A= acsin β, and ac( a + c b)sin β ac( a + c b)sinβ = =. ( a+ c) b a + c b + ac
4 4 Fom te law of cosines we ave + = cos, and te last expession a c b ac β becomes ac( a + c b)sin β ( a + c b)sinβ = = ac cosβ + ac (+ cos β). But sin β β = tan, and te expession above becomes (3). Relations () and (4) + cos β follow fom te symmety of te tiangle. Teoem A: Te inadius of a igt tiangle wit legs a and b and ypotenuse c is (5) a+ b c =. Poof: Since we ave a igt tiangle, angle γ = π /. Te esult (5) follows immediately fom (4). Te following teoems follow easily fom Teoems and A. Recall tat a Pytagoean tiangle is a igt tiangle wit all tee sides integes, and a Heonian tiangle as all tee sides and aea given by integes. A pimitive Pytagoean tiangle is one in wic all tee sides ave no common facto. Teoem 3: Te inadius of a Pytagoean tiangle is an intege. Poof: Tis follows fom (5) and te fact tat eite one side, o all tee sides of te Pytagoean tiangle ae even numbes. Teoem 4: Te inadius of a Heonian tiangle is a ational numbe. Poof: Tis follows immediately fom (). If te inadius of a Heonian tiangle is te ational numbe = m/ n, we can multiply all tee sides by n to obtain a simila Heonian tiangle wit intege inadius = m.
5 5 Teoem 5: Te tigonometic functions of te inteio angles αβand, γ of a Heonian tiangle ae all ational numbes. Also tan ( α / ), tan ( β / ) and tan ( / ) γ ae ational. Poof: Refe to Figue. Fom (), (3) and (4) it follows tat te tangents of te alf tan( α /) angles ae all ational. Since tanα =, it follows tat tanα is ational. In tan ( α /) te same way tan β and tanγ ae ational. Fom te law of cosines we get b + c a cosα =, and tus cosα is ational. Te same must be tue fo cos β and bc cosγ. Since te tangents and cosines of all tee angles of a Heonian tiangle ae ational, it follows tat te emaining tigonometic functions of tese angles ae ational. We can constuct a Heonian tiangle by joining togete two Pytagoean tiangles using te following steps:. Select any two Pytagoean tiangles and place tem as sown in Figue. Figue. Multiplying eac tiangle by appopiate integes we obtain two new simila Pytagoean tiangles wit identical altitudes as sown in Figue 3.
6 c α α b b c 6 β a a β Figue 3 3. Now place te tiangles side by side to fom a new Heonian tiangle as sown in Figue 4. Notice tat we ave set = b = b. Figue 4 By evesing te steps we can stat wit any Heonian tiangle, and decompose it into two Pytagoean tiangles. To see tis look at Figue 4 and notice tat te igt tiangle on te left side as altitude = csin β and base a = ccos β. Now c is an intege, and by Teoem 5, sin β and cos β ae ational numbes. Tus te igt tiangle on te left side as ational sides and tus is simila to a Pytagoean tiangle. Te same is tue of te tiangle on te igt side. c a= a+ a Fo te moment, let te two tiangles in Figue 3 be any two igt tiangles (not necessaily Pytagoean) wit identical altitudes ( = b = b). Te following teoem c β β
7 7 elates te inadius of tei union (sown in Figue 4) wit te inadii and of te individual igt tiangles. Teoem 6: Wit te notation of Figues 3 and 4 we ave (6) ( ) Poof: Fom () we ave α = tan, wee α = α+ α. c+ c ( a+ a) α = tan c a + c a α = tan. Adding and subtacting b and b we get b + b + ( c a b) + ( c a b ) α tan =. a+ b c Using (5) fo te inadius of a igt tiangle we see tat = and a + b c =. Also, poved. = b = b and te above expession becomes (6). Te teoem is Now conside te case in wic te two igt tiangles used above ae conguent. Tei esulting union is an isosceles tiangle wit c= c = c, a = a, α = α = α /, and =. Te inadius of tis isosceles tiangle can now be calculated fom (6) as a = ( ),
8 8 α a wee we ave used tan = tanα =. We can use (5) to get = a+ c. Replacing ` in te above expession we get ( a c ) a = ( c ) a, and we ave poved te following teoem. = ( + ) a. Tis simplifies to Teoem 7: Given an isosceles tiangle wit base a, altitude and two sides equal to c, te inadius is given by ( c a) a (7) =. Teoem 8: Suppose a Heonian isosceles tiangle is fomed by te union of two identical pimitive Pytagoean tiangles wit legs a and and ypotenuse c. Ten te incicle cannot ave intege adius. Poof: In a pimitive Pytagoean tiangle, te sides a,, and c ave no common facto. If te inadius wee an intege, ten (7) sows tat divides ( c a). If divides ( c a), c a ten = m, an intege. But c a c a =, and so =. Tis means tat c + a c+ a = m wic is impossible since c+ a >. Refeences [] Fine, Ia and Osle, Tomas, Te emakable incicle of a tiangle., Matematics and Compute Education, 35(00), pp [] Jonson, R. A., Moden Geomety: An Elementay Teatise on te Geomety of te Tiangle and te Cicle, Hougton, Mifflin, Boston, MA., 99.
9 9 [3] Long, Calvin T., Elementay Intoduction to Numbe Teoy, Heat & Co., 967, pp [4] Sasty, K. R. S., Heon tiangles: An incente pespective, Matematics Magazine, 73(000), pp
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