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DO NOW: TAKE OUT CALCULATOR DEFINE CEVIAN Aim: To understand Stewart’s Theorem A cevian is a straight line that passes through a vertex of a triangle or tetrahedron and intersects the opposite side or face.

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Stewart’s Theorem Matthew Stewart (1717–1785) published this theorem as Proposition II on page 2 in his 1746 book. It is said that Archimedes used the theorem though (300 B.C.) In geometry, Stewart's theorem yields a relation between a lengths of the sides of the triangle and the length of a cevian of the triangle.

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b 2 m+c 2 n=a(d 2 +mn) a bc d nm

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N=5 A B C D5 34 13 17 b 2 mc+ 2 n=a(d 2 +mn) 25 20 5 * A=25 B=34 C=13 D=17 M=20

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This is also often written, a form which invites mnemonic memorization, e.g. "A man and his dad put a bomb in the sink." b 2 m+c 2 n=amn+d 2 aman+dad=bmb+cnc

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A = 16 cm N = 8 cm M = 8 cm D = 15 cm C = 17 cm B = 17 cm D

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Quinn RachelSchoolPenny Penny, Quinn, and Rachel all go to the same school. Penny lives 5.5399 miles from Quinn, who lives 5.0803 miles from Rachel. If the school is between Penny and Rachel and Penny is 5.7073 miles away from it and Rachel is 2.2927 miles away from it in the opposite direction, how far is Quinn from the school? 5.7073 5.0803 2.2927 x 5.5399

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b 2 m+c 2 n=a(d 2 +mn ) Quinn RachelPenny 5.7073 5.0803 2.2927 x 5.5399 School 3.75

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A = 18 B = 17 C = 16 M = 8 N = 10 D = 13.8 17 16 810

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MEDIAN ANGLE BISECTOR ALTITUDE If the cevian happens to be a median, angle bisector, or altitude the following formulas can be used

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1. HTTP://EASYCALCULATION.COM/THEORE MS/STEWART-OR-APOLLONIUS.PHP HTTP://EASYCALCULATION.COM/THEORE MS/STEWART-OR-APOLLONIUS.PHP 2. HTTP://MATH.STACKEXCHANGE.COM/QUE STIONS/46616/PROVING-STEWARTS- THEOREM-WITHOUT-TRIG HTTP://MATH.STACKEXCHANGE.COM/QUE STIONS/46616/PROVING-STEWARTS- THEOREM-WITHOUT-TRIG 3. HTTP://WWW.ANSWERS.COM/TOPIC/CEVI AN-MATHEMATICS HTTP://WWW.ANSWERS.COM/TOPIC/CEVI AN-MATHEMATICS 4. HTTP://WWW.ARTOFPROBLEMSOLVING.C OM/WIKI/INDEX.PHP/STEWART'S_THEOR EM HTTP://WWW.ARTOFPROBLEMSOLVING.C OM/WIKI/INDEX.PHP/STEWART'S_THEOR EM 5. HTTP://WWW.MS.UKY.EDU/~DROYSTER/C OURSES/FALL11/MA341/CLASSNOTES/LEC TURE%2013%20HANDOUTS.PDF HTTP://WWW.MS.UKY.EDU/~DROYSTER/C OURSES/FALL11/MA341/CLASSNOTES/LEC TURE%2013%20HANDOUTS.PDF Sources

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Honors Geometry Section 8. 5

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