Mathematics Dictionary Dr. K. G. Shih

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Ex-center

Subjects

Symbol Defintion Sqr(x) = Square root of x Q01 | - What is ex-center of a triangle ? Q02 | - Prove that bisectors of angles of triangle are concurrent Q03 | - Ex-circle tangents the sides of triangle Q04 | - Locus of ex-center Q05 | - Using coordinate geometry Prove that bisectors of angles of triangle concurrent Q06 | - Coordinates of ex-center Q07 | - Q08 | - Q09 | - Q10 | -

Answers

Q01. What is in-center Definition Bisectors of one internal angle A and external angles B and C They are concurrent at a point I The point is called ex-center A triangle has three ex-centers What is concurrent ? Three lines meet at one point is called concurrent What is bisector of an angle ? A line biscets an angle equally which is bisector of an angle Bisector theory of angle point on bisector of an angle has same distance from the two rays If point between the rays of angle has same distance, point is on bisector diagram Diagram of geometry Program 03 02 Go to Begin Q02. bisectors of angles of triangle are concurrent Construction Draw triangle ABC Let bisector of external angles B and C meet at I Join A and I as a line We want to prove that AI is bisector of angle A Proof Draw IF perpendicular to AB, ID perpendicular to BC IF = ID (bisector theory) Draw ID perpendicular to BC, IE perpendicular to CA IE = ID (bisector theory) Hence IE = IF Hence AI is bisector of angle A (bisector theory) Hence bisectors of angles of triangle are concurrent Go to Begin Q03. In-circle tangents the sides of triangle In Q02, ID = IE = IF = in-radius = r1 Hence use I as center and r1 as radius to make a circle The circle will tangent the sides of triangle Go to Begin Q04. Locus of circum-center Conditions Let A and B are fixed and C is moving with angle ACB = constant Find locus of ex-center Diagram Diagram of geometry Program 10 02 Proof We can proof that Angle AIB = pi/2 - A/2 - B/2 Angle AIB = pi/2 - C/2 Since angle AIB is constant and point A and point B are fixed Hence point I will move along arc AIB (circum-circle of tringle AIB) Go to Begin Q05. Using coordinate geometry Prove bisectors of angles of triangle concurrent Reference Coordinate geometry Program 11 ?? Go to Begin Q06. Find coordinate of ex-center Use tangent theory We can prove that AF = AE = s = length of tangent from A to F or A to E Where s = (a+b+c)/2 and a,b,c are lenght of sides Find coordinate of in-center Method 1 Let AB be in horizontal direction and A be the origin The center I(xi,yi) xi = s yi = r1*tan(A/2) Method 2 Find equations of two bisectors using slope and point Find intersections of two equations The intersection is the ex-center Go to Begin Q07. Answer Go to Begin Q08. Answer Go to Begin Q09. Answer Go to Begin Q10. Answer Go to Begin

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