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.11).If, instead, the exterior angle trisectors are used then another equilateraltriangle is formed (the second Morley triangle) and, moreover, the intersectionsof the sides of this triangle with the external trisectors form three additionalequilateral triangles [322].Property 12 (Fermat-Torricelli Problem).In 1629, Fermat challengedTorricelli to find a point whose total sum of distances from the vertices of atriangle is a minimum [224].I.e., determine a point X (Fermat point) in agiven triangle ABC such that the sum XA + XB + XC is a minimum [178].If the angle at one vertex is greater than or equal to 120æ% then the Fermatpoint coincides with this vertex.Otherwise, the Fermat point coincides withthe so-called (inner) isogonic center (X in Figure 2.12) which may be found byconstructing outward pointing equilateral triangles on the sides of "ABC and 36 Mathematical PropertiesFigure 2.12: Fermat Pointconnecting each vertex of the original triangle to the new vertex of the oppositeequilateral triangle.These three segments are of equal length and intersect atthe isogonic center where they are inclined at 60æ% to one another.At theisogonic center, each side of the original triangle subtends an angle of 120æ%.Also, the isogonic center lies at the common intersection of the circumcircles ofthe three equilateral triangles [322].The algebraic sum of the distances fromthe isogonic center to the vertices of the triangle equals the length of the equalsegments from the latter to the opposite vertices of the equilateral triangles[189].Property 13 (Largest Circumscribed Equilateral Triangle).If we con-nect the isogonic center of an arbitrary triangle with its vertices and draw linesthrough the latter perpendicular to the connectors then these lines intersect toform the largest equilateral triangle circumscribing the given triangle [189].This is the antipedal triangle of the isogonic center with respect to the giventriangle.Property 14 (Napoleon s Theorem).On each side of an arbitrary triangle,construct an equilateral triangle pointing outwards.The centers of these threetriangles form an equilateral triangle [178] called the outer Napoleon triangle(Figure 2.13(a)). Mathematical Properties 37(a) (b) (c)Figure 2.13: Napoleon s Theorem [322, 178, 245]If, instead, the three equilateral triangles point inwards then another equi-lateral triangle is formed (the inner Napoleon triangle) and, moreover, the twoNapoleon triangles share the same center with the original equilateral triangleand the difference in their areas is equal to the area of the original triangle[322].(See [71] for a most interesting conjectured provenance for this theorem.)Also, the lines joining a vertex of either Napoleon triangle with the remote ver-tex of the original triangle are concurrent (Figure 2.13(b)) [245].Finally, thelines joining each vertex of either Napoleon triangle to the new vertex of thecorresponding equilateral triangle drawn on each side of the original equilat-eral triangle are conccurrent and, moreover, the point of concurrency is thecircumcenter of the original equilateral triangle (Figure 2.13(c)) [245].(b)(a)Figure 2.14: Parallelogram Properties [245].Property 15 (Parallelogram Properties).With reference to Figure 2.14(a),equilateral triangles BCE and CDF are constructed on sides BC and CD, re-spectively, of a parallelogram ABCD.Since triangles BEA, CEF and DAFare congruent, triangle AEF is equilateral [245].With reference to Figure 38 Mathematical Properties2.14(b) left/right, construct equilateral triangles pointing outwards/inwards onthe sides of an oriented parallelogram ABCD giving parallelogram XY ZW.Then, if inward/outward pointing equilateral triangles are drawn on the sidesof oriented parallelogram XY ZW , the resulting parallelogram is just ABCDagain [182].(b)(a)Figure 2.15: (a) Isodynamic (Apollonius) Points.(b) Pedal Triangle of FirstIsodynamic PointProperty 16 (Pedal Triangles of Isodynamic Points).With referenceto "ABC of Figure 2.15(a), let U and V be the points on BC met by theinterior and exterior bisectors of "A.The circle having diameter UV is calledthe A-Apollonian circle [6] [ Pobierz caÅ‚ość w formacie PDF ]
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.11).If, instead, the exterior angle trisectors are used then another equilateraltriangle is formed (the second Morley triangle) and, moreover, the intersectionsof the sides of this triangle with the external trisectors form three additionalequilateral triangles [322].Property 12 (Fermat-Torricelli Problem).In 1629, Fermat challengedTorricelli to find a point whose total sum of distances from the vertices of atriangle is a minimum [224].I.e., determine a point X (Fermat point) in agiven triangle ABC such that the sum XA + XB + XC is a minimum [178].If the angle at one vertex is greater than or equal to 120æ% then the Fermatpoint coincides with this vertex.Otherwise, the Fermat point coincides withthe so-called (inner) isogonic center (X in Figure 2.12) which may be found byconstructing outward pointing equilateral triangles on the sides of "ABC and 36 Mathematical PropertiesFigure 2.12: Fermat Pointconnecting each vertex of the original triangle to the new vertex of the oppositeequilateral triangle.These three segments are of equal length and intersect atthe isogonic center where they are inclined at 60æ% to one another.At theisogonic center, each side of the original triangle subtends an angle of 120æ%.Also, the isogonic center lies at the common intersection of the circumcircles ofthe three equilateral triangles [322].The algebraic sum of the distances fromthe isogonic center to the vertices of the triangle equals the length of the equalsegments from the latter to the opposite vertices of the equilateral triangles[189].Property 13 (Largest Circumscribed Equilateral Triangle).If we con-nect the isogonic center of an arbitrary triangle with its vertices and draw linesthrough the latter perpendicular to the connectors then these lines intersect toform the largest equilateral triangle circumscribing the given triangle [189].This is the antipedal triangle of the isogonic center with respect to the giventriangle.Property 14 (Napoleon s Theorem).On each side of an arbitrary triangle,construct an equilateral triangle pointing outwards.The centers of these threetriangles form an equilateral triangle [178] called the outer Napoleon triangle(Figure 2.13(a)). Mathematical Properties 37(a) (b) (c)Figure 2.13: Napoleon s Theorem [322, 178, 245]If, instead, the three equilateral triangles point inwards then another equi-lateral triangle is formed (the inner Napoleon triangle) and, moreover, the twoNapoleon triangles share the same center with the original equilateral triangleand the difference in their areas is equal to the area of the original triangle[322].(See [71] for a most interesting conjectured provenance for this theorem.)Also, the lines joining a vertex of either Napoleon triangle with the remote ver-tex of the original triangle are concurrent (Figure 2.13(b)) [245].Finally, thelines joining each vertex of either Napoleon triangle to the new vertex of thecorresponding equilateral triangle drawn on each side of the original equilat-eral triangle are conccurrent and, moreover, the point of concurrency is thecircumcenter of the original equilateral triangle (Figure 2.13(c)) [245].(b)(a)Figure 2.14: Parallelogram Properties [245].Property 15 (Parallelogram Properties).With reference to Figure 2.14(a),equilateral triangles BCE and CDF are constructed on sides BC and CD, re-spectively, of a parallelogram ABCD.Since triangles BEA, CEF and DAFare congruent, triangle AEF is equilateral [245].With reference to Figure 38 Mathematical Properties2.14(b) left/right, construct equilateral triangles pointing outwards/inwards onthe sides of an oriented parallelogram ABCD giving parallelogram XY ZW.Then, if inward/outward pointing equilateral triangles are drawn on the sidesof oriented parallelogram XY ZW , the resulting parallelogram is just ABCDagain [182].(b)(a)Figure 2.15: (a) Isodynamic (Apollonius) Points.(b) Pedal Triangle of FirstIsodynamic PointProperty 16 (Pedal Triangles of Isodynamic Points).With referenceto "ABC of Figure 2.15(a), let U and V be the points on BC met by theinterior and exterior bisectors of "A.The circle having diameter UV is calledthe A-Apollonian circle [6] [ Pobierz caÅ‚ość w formacie PDF ]