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.Property 44 (Jung s Theorem).Let d be the (finite) diameter of a planarset and let r be the radius of its smallest enclosing circle.Then, [249]d"r d".3Since the circumcircle is the smallest enclosing circle for an equilateral triangle(Figure 2.32), this bound cannot be diminished.Property 45 (Isoperimetric Theorem for Triangles).Among trianglesof a given perimeter, the equilateral triangle has the largest area [191].Equiv-alently, among all triangles of a given area, the equilateral triangle has theshortest perimeter [228].Property 46 (A Triangle Inequality).If A is the area and L the perimeterof a triangle then"A d" 3L2/36,with equality if and only if the triangle is equilateral [228].Property 47 (Euler s Inequality).If r and R are the radii of the inscribedand circumscribed circles of a triangle thenR e" 2r,with equality if and only if the triangle is equilateral [191, 228].Mathematical Properties 53Property 48 (Erdös-Mordell Inequality).Let R1, R2, R3 be the distancesto the three vertices of a triangle from any interior point P.Let r1, r2, r3 bethe distances from P to the three sides.ThenR1 + R2 + R3 e" 2(r1 + r2 + r3),with equality if and only if the triangle is equilateral and P is its centroid[191, 228].Property 49 (Blundon s Inequality).In any triangle ABC with circum-radius R, inradius r and semi-perimeter Ã, we have that"à d" 2R + (3 3 - 4)r,with equality if and only if ABC is equilateral [26].Property 50 (Garfunkel-Bankoff Inequality).If Ai (i = 1, 2, 3) are theangles of an arbitrary triangle, then we have3 3Ai Aitan2 e" 2 - 8 sin ,2 2i=1 i=1with equality if and only if ABC is equilateral [331].Property 51 (Improved Leunberger Inequality).If si (i = 1, 2, 3) arethe sides of an arbitrary triangle with circumradius R and inradius r, then wehave"31 25Rr - 2r2e" ,si 4Rri=1with equality if and only if ABC is equilateral [331].Figure 2.33: Shortest Bisecting Path [161]Property 52 (Shortest Bisecting Path).The shortest path across an equi-lateral triangle of side s which bisects its area is given by a circular arc withcenter at a vertex and with radius chosen to bisect the area (Figure 2.33) [228].54 Mathematical PropertiesFigure 2.34: Smallest Inscribed Triangle [57, 62]"3 3This radius is equal to s · [161] so that the circular arc has length4À.673.s which is much shorter than either the.707.s length of the parallelbisector or the.866.s length of the altitude.Property 53 (Smallest Inscribed Triangle).The problem of finding thetriangle of minimum perimeter inscribed in a given acute triangle [62] wasposed by Giulio Fagnano and solved using calculus by his son Giovanni Fagnanoin 1775 [224].(An inscribed triangle being one with a vertex on each side ofthe given triangle.) The solution is given by the orthic/pedal triangle of thegiven acute triangle (Figure 2.34 left).Later, H.A.Schwarz provided a geometric proof using mirror reflections[57].Call the process illustrated on the left of Figure 2.34 the pedal mapping.Then, the unique fixed point of the pedal mapping is the equilateral triangle[194].That is, the equilateral triangle is the only triangle that maintains itsform under the pedal mapping.Also, the equilateral triangle is the only trian-gle for which successive pedal iterates are all acute [205].Finally, the maximalratio of the perimeter of the pedal triangle to the perimeter of the given acutetriangle is 1/2 and the unique maximizer is given by the equilateral triangle[158].For a given equilateral triangle, the orthic/pedal triangle coincides withthe medial triangle which is itself equilateral (Figure 2.34 right).Property 54 (Closed Light Paths [57]).The walls of an equilateral tri-angular room are mirrored.If a light beam emanates from the midpoint of awall at an angle of 60æ%, it is reflected twice and returns to its point of originby following a path along the pedal triangle (see Figure 2.34 right) of the room.If it originates from any other point along the boundary (exclusive of corners)at an angle of 60æ%, it is reflected five times and returns to its point of originby following a path everywhere parallel to a wall (Figure 2.35) [57] [ Pobierz caÅ‚ość w formacie PDF ]