% , E0 capmnRU F hUjxjlh@nPAiP`}4iiPqZj(0is8ij_+0i+h hwWritten by: Ken Koedinger, School of Computer Science, Carnegie Mellon University, Pittsburgh, PA 15213 (412) 268-7667 Email: koedinger@cs.cmu.edududu$`UEhDteaseo  Z?C jxjlh@nPAiP`}4iiPqZj(0is8ij_+0i+hDEkB. Strategy: 1. Move point F until that it belonged to John Anderson. Who does it belong to (primarily, if such), > and what i1?Ueaseo jxjlh@nPPiP`}4iiPqZj(0is8ij_+0i+h 8=A. Problem formulation: Given CD and DE, maximize angle DFE. eg`W umHX pD0(HHBBIN jxjlh@nPPiP`}4iiPqZj(0is8ij_+0i+h+ BRObservation: DFE seems maximal just when triangle FCE is similar to triangle DCF.F.D?`FUO/./+Hm h=_`.@gS@gS@gS@XpSpS5D <B'! jxjlh@nPPiP`}4iiPqZj(0is8ij_+0i+h  *@TewxDFE is maximized when: 1. Angle FGC is 45 (where G midpoint DE) 2. Ratio of FDC and FEC is maximized 3. DF bisects CFE 4. Similar triangles*"  &5PZl1X} jxjlh@nPPiP`}4iiPqZj(0is8ij_+0i+h +ZntIf the above observation is true, then DFE should be maximized when CF/CE = CD/CF or CF = sqrt( (CD+DE) * CD) Check:,xkGkGll0,xkGkGl)'/BgBgjxjlh@nPjiP`}4iiPqZj(0is8ij_+0i+hB0AB0C?*/?O̖jxjlh@n PCiP`}4iiPqZj(0is8ij_+0i+hB0C̀  *#/?jxjlh@n PFiP`}4iiPqZj(0is8ij_+0i+hB0C 'jxjlh@nPk1iP`}4iiPqZj(0is8ij_+0i+hCC̀CC̀? )/tj jxjlh@nPt1iP`}4iiPqZj(0is8ij_+0i+hB0C̀B0C? 8 jxjlh@nP11iP`}4iiPqZj(0is8ij_+0i+h FC = o.MMTpp!!|h@p(#o-!`Rp0pP.Op@U!`^zounRounh@p m`>T4pDistance(C to F) = q.B-0Ha:aRecs():4TpTp4aĸpR@4|P|HqbP4pq|2qXa2q|23_X):|Recs():qIqzq|2s(): qIbP|LL pr`qr@Prrr JKwr`rc8`Pyr`Pհw  `|WDjtPdc~s y~>ayjxjlh@n PD1iP`}4iiPqZj(0is8ij_+0i+hBC̀ =]0jxjlh@n PE1iP`}4iiPqZj(0is8ij_+0i+hCeC̀ x jxjlh@nPm2iP`}4iiPqZj(0is8ij_+0i+hBC̀CeC̀?)~ jxjlh@nPn2iP`}4iiPqZj(0is8ij_+0i+hBC̀B0C?) jxjlh@nPp2iP`}4iiPqZj(0is8ij_+0i+hB0CCeC̀?N&] jxjlh@nP12iP`}4iiPqZj(0is8ij_+0i+h  m{!:A}DFE = .MMTpp!!|h@p(#o-!`Rp0pP.Op@U!`^zounRounh@p m`>T4p Angle(DFE) = F) = q.B-0Ha:aRecs():4TpTp4aĸpR@4|P|HqbP4pq|2qXa2q|23_X):|Recs():qIqzq|2s(): qIbP|LL pr`qr@Prrr JKwr`rc8`Pyr`Pհw  `|WDjtPdc~s< jxjlh@nP32iP`}4iiPqZj(0is8ij_+0i+h  m{!:A}FDC = .MMTpp!!|h@p(#o-!`Rp0pP.Op@U!`^zounRounh@p m`>T4p Angle(FDC) = F) = q.B-0Ha:aRecs():4TpTp4aĸpR@4|P|HqbP4pq|2qXa2q|23_X):|Recs():qIqzq|2s(): qIbP|LL pr`qr@Prrr JKwr`rc8`Pyr`Pհw  `|WDjtPdc~s Iv  jxjlh@nP42iP`}4iiPqZj(0is8ij_+0i+h  m{!:A}FEC = .MMTpp!!|h@p(#o-!`Rp0pP.Op@U!`^zounRounh@p m`>T4p Angle(FEC) = F) = q.B-0Ha:aRecs():4TpTp4aĸpR@4|P|HqbP4pq|2qXa2q|23_X):|Recs():qIqzq|2s(): qIbP|LL pr`qr@Prrr JKwr`rc8`Pyr`Pհw  `|WDjtPdc~s M jxjlh@nP62iP`}4iiPqZj(0is8ij_+0i+h  m{!:A}EFD = .MMTpp!!|h@p(#o-!`Rp0pP.Op@U!`^zounRounh@p m`>T4p Angle(EFD) = F) = q.B-0Ha:aRecs():4TpTp4aĸpR@4|P|HqbP4pq|2qXa2q|23_X):|Recs():qIqzq|2s(): qIbP|LL pr`qr@Prrr JKwr`rc8`Pyr`Pհw  `|WDjtPdc~s> jxjlh@nP72iP`}4iiPqZj(0is8ij_+0i+h  m{!:A}DFC = .MMTpp!!|h@p(#o-!`Rp0pP.Op@U!`^zounRounh@p m`>T4p Angle(DFC) = F) = q.B-0Ha:aRecs():4TpTp4aĸpR@4|P|HqbP4pq|2qXa2q|23_X):|Recs():qIqzq|2s(): qIbP|LL pr`qr@Prrr JKwr`rc8`Pyr`Pհw  `|WDjtPdc~s )p jxjlh@nPr2iP`}4iiPqZj(0is8ij_+0i+hB0C̀CeC̀? )~"XO jxjlh@nPs2iP`}4iiPqZj(0is8ij_+0i+hBC̀B0C̀? 2+, jxjlh@nP12iP`}4iiPqZj(0is8ij_+0i+h DC = }DFC = .MMTpp!!|h@p(#o-!`Rp0pP.Op@U!`^zounRounh@p m`>T4pDistance(C to D) = q.B-0Ha:aRecs():4TpTp4aĸpR@4|P|HqbP4pq|2qXa2q|23_X):|Recs():qIqzq|2s(): qIbP|LL pr`qr@Prrr JKwr`rc8`Pyr`Pհw  `|WDjtPdc~s 9Y  jxjlh@nP13iP`}4iiPqZj(0is8ij_+0i+h ED = }DFC = .MMTpp!!|h@p(#o-!`Rp0pP.Op@U!`^zounRounh@p m`>T4pDistance(D to E) = q.B-0Ha:aRecs():4TpTp4aĸpR@4|P|HqbP4pq|2qXa2q|23_X):|Recs():qIqzq|2s(): qIbP|LL pr`qr@Prrr JKwr`rc8`Pyr`Pհw  `|WDjtPdc~sY jxjlh@nPG4iP`}4iiPqZj(0is8ij_+0i+hC0C̀ |P jxjlh@nP54iP`}4iiPqZj(0is8ij_+0i+h  {D:m{!:A}FDC}{m{!:A}FEC} = Tpp!!|h@p(#o-!`Rp0pP.Op@U!`^zounRounh@p m`>T4pAngle(FDC)/Angle(FEC) = B-0Ha:aRecs():4TpTp4aĸpR@4|P|HqbP4pq|2qXa2q|23_X):|Recs():qIqzq|2s(): qIbP|LL pr`qr@Prrr JKwr`rc8`Pyr`Pհw  `|WDjtPdc~set` jxjlh@nP84iP`}4iiPqZj(0is8ij_+0i+h  {D:m {S:CF}}{m {S:CE}} = = Tpp!!|h@p(#o-!`Rp0pP.Op@U!`^zounRounh@p m`>T4p Ratio(t/r) = gle(FEC) = B-0Ha:aRecs():4TpTp4aĸpR@4|P|HqbP4pq|2qXa2q|23_X):|Recs():qIqzq|2s(): qIbP|LL pr`qr@Prrr JKwr`rc8`Pyr`Pհw  `|WDjtPdc~sWf_ jxjlh@nP94iP`}4iiPqZj(0is8ij_+0i+h  {D:m {S:DC}}{m {S:CF}} = = Tpp!!|h@p(#o-!`Rp0pP.Op@U!`^zounRounh@p m`>T4p Ratio(s/t) = gle(FEC) = B-0Ha:aRecs():4TpTp4aĸpR@4|P|HqbP4pq|2qXa2q|23_X):|Recs():qIqzq|2s(): qIbP|LL pr`qr@Prrr JKwr`rc8`Pyr`Pհw  `|WDjtPdc~s9&P\ jxjlh@nP14iP`}4iiPqZj(0is8ij_+0i+h  {V:{(:DC + ED}{!:*}DC} = = Tpp!!|h@p(#o-!`Rp0pP.Op@U!`^zounRounh@p m`>T4p=Sqrt[(Distance(C to D)+Distance(D to E))*Distance(C to D)] = s():4TpTp4aĸpR@4|P|HqbP4pq|2qXa2q|23_X):|Recs():qIqzq|2s(): qIbP|LL pr`qr@Prrr JKwr`rc8`Pyr`Pհw  `|WDjtPdc~s 5J jxjlh@nP24iP`}4iiPqZj(0is8ij_+0i+h  m{!:A}FGC = D}{!:*}DC} = = Tpp!!|h@p(#o-!`Rp0pP.Op@U!`^zounRounh@p m`>T4p Angle(FGC) = e(C to D)+Distance(D to E))*Distance(C to D)] = s():4TpTp4aĸpR@4|P|HqbP4pq|2qXa2q|23_X):|Recs():qIqzq|2s(): qIbP|LL pr`qr@Prrr JKwr`rc8`Pyr`Pհw  `|WDjtPdc~s