❙✉✐❞✲❆❢r✐❦❛❛♥s❡ ❲✐s❦✉♥❞❡✲❖❧✐♠♣✐❛❞❡
❉❡r❞❡ ❘♦♥❞❡ ✷✵✵✸
❚②❞✿ ✹ ✉✉r
✶✳ ❏② ❤❡t ✈②❢ st✉❦❦❡ ♣❛♣✐❡r✳ ❏② ❦✐❡s ❡❡♥ ♦❢ ♠❡❡r ❞❛❛r✈❛♥ ❡♥ s♥② ❡❧❦❡❡♥ ♦♣ ✐♥ ✈②❢ ❦❧❡✐♥❡r st✉❦❦❡✳ ◆♦✉ ♥❡❡♠ ❥② ❡❡♥ ♦❢ ♠❡❡r st✉❦❦❡ ✈❛♥ ❤✐❡r❞✐❡ ❦❧♦♠♣ ❡♥ s♥② ❡❧❦❡❡♥ ♦♣ ✐♥ ✈②❢ ❦❧❡✐♥❡r st✉❦❦❡✳ ❊♥s♦✈♦♦rts✳ ❇❡✇②s ❞❛t ❥② ♥♦♦✐t ✷✵✵✸ st✉❦❦❡ s❛❧ ❤❫❡ ♥✐❡✳
✷✳ ●❡❣❡❡ ✬♥ ♣❛r❛❧❧❡❧♦❣r❛♠ ABCD, ✈❡r❜✐♥❞ A ♠❡t ❞✐❡ ♠✐❞❞❡❧♣✉♥t❡ E ❡♥ F
✈❛♥ ❞✐❡ t❡❡♥♦♦rst❛❛♥❞❡ s②❡ BC ❡♥ CD. AE ❡♥ AF s♥② ❞✐❡ ❞✐❛❣♦♥❛❛❧ BD
✐♥M❡♥ N. ❇❡✇②s ❞❛tM ❡♥ N✈✐rBD ✐♥ ❞r✐❡ ❣❡❧②❦❡ ❞❡❧❡ ✈❡r❞❡❡❧✳
✸✳ ❉✐❡ ❡❡rst❡ ✈✐❡r s②❢❡rs ✈❛♥ ✬♥ s❡❦❡r❡ ♣♦s✐t✐❡✇❡ ❤❡❡❧t❛❧n✐s ✶✶✸✼✳ ❇❡✇②s ❞❛t
❞✐❡ s②❢❡rs ✈❛♥ n s♦ ❣❡s❦♦♠♠❡❧ ❦❛♥ ✇♦r❞ ❞❛t ❞✐❡ ♥✉✇❡ ❣❡t❛❧ ❞❡❡❧❜❛❛r ✐s
❞❡✉r ✼✳
✹✳ ■♥ ✬♥ ❣❡❣❡✇❡ ✈②❢❤♦❡❦ABCDE ❤❡t ❞✐❡ ❞r✐❡❤♦❡❦❡ABC, BCD, CDE, DEA
❡♥EAB❛❧♠❛❧ ❞✐❡s❡❧❢❞❡ ❛r❡❛✳ ❉✐❡ ❧②♥❡AC ❡♥ADs♥②BE✐♥ ❞✐❡ ♣✉♥t❡M
❡♥N. ❇❡✇②s ❞❛tBM=EN.
✺✳ ❇❡✇②s ❞❛t ❞✐❡ s♦♠ ✈❛♥ ❞✐❡ ❦✇❛❞r❛t❡ ✈❛♥ t✇❡❡ ♦♣❡❡♥✈♦❧❣❡♥❞❡ ♣♦s✐t✐❡✇❡ ❤❡❡❧t❛❧❧❡ ♥✐❡ ❣❡❧②❦ ❦❛♥ ✇❡❡s ❛❛♥ ❞✐❡ s♦♠ ✈❛♥ ❞✐❡ ✈✐❡r❞❡♠❛❣t❡ ✈❛♥ t✇❡❡ ♦♣❡❡♥✈♦❧❣❡♥❞❡ ♣♦s✐t✐❡✇❡ ❤❡❡❧t❛❧❧❡ ♥✐❡✳
✻✳ ■♥△ABC✐s ❞✐❡ s♦♠ ✈❛♥ ❞✐❡ s②❡2s❡♥ ❞✐❡ str❛❛❧ ✈❛♥ ❞✐❡ ✐♥❣❡s❦r❡✇❡ s✐r❦❡❧ r.❉r✐❡ ❤❛❧❢s✐r❦❡❧s ♠❡t ♠✐❞❞❡❧❧②♥❡AB, BC ❡♥CA ✇♦r❞ ❛❛♥ ❞✐❡ ❜✉✐t❡❦❛♥t
✈❛♥ABC❣❡t❡❦❡♥✳ ✬♥ ❙✐r❦❡❧ ✈❛♥ str❛❛❧tr❛❛❦ ❛❛♥ ❛❧ ❞r✐❡ ❤❛❧❢s✐r❦❡❧s✳ ❇❡✇②s
❞❛t
s 2 < t≤
s
2 + 1−
√
3 2
!
r.
❙♦✉t❤ ❆❢r✐❝❛♥ ▼❛t❤❡♠❛t✐❝❛❧ ❖❧②♠♣✐❛❞
❚❤✐r❞ ❘♦✉♥❞ ✷✵✵✸
❚✐♠❡✿ ✹ ❤♦✉rs
✶✳ ❨♦✉ ❤❛✈❡ ☞✈❡ ♣✐❡❝❡s ♦❢ ♣❛♣❡r✳ ❨♦✉ ♣✐❝❦ ♦♥❡ ♦r ♠♦r❡ ♦❢ t❤❡♠ ❛♥❞ ❝✉t ❡❛❝❤ ♦❢ t❤❡♠ ✐♥t♦ ☞✈❡ s♠❛❧❧❡r ♣✐❡❝❡s✳ ◆♦✇ ②♦✉ t❛❦❡ ♦♥❡ ♦r ♠♦r❡ ♦❢ t❤❡ ♣✐❡❝❡s ❢r♦♠ t❤✐s ❧♦t ❛♥❞ ❝✉t ❡❛❝❤ ♦❢ t❤❡s❡ ✐♥t♦ ☞✈❡ s♠❛❧❧❡r ♣✐❡❝❡s✳ ❆♥❞ s♦ ♦♥✳ Pr♦✈❡ t❤❛t ②♦✉ ✇✐❧❧ ♥❡✈❡r ❤❛✈❡ ✷✵✵✸ ♣✐❡❝❡s✳
✷✳ ●✐✈❡♥ ❛ ♣❛r❛❧❧❡❧♦❣r❛♠ ABCD, ❥♦✐♥ A t♦ t❤❡ ♠✐❞♣♦✐♥ts E ❛♥❞ F ♦❢ t❤❡
♦♣♣♦s✐t❡ s✐❞❡s BC ❛♥❞ CD. AE ❛♥❞ AF ✐♥t❡rs❡❝t t❤❡ ❞✐❛❣♦♥❛❧ BD ✐♥ M
❛♥❞N. Pr♦✈❡ t❤❛tM ❛♥❞ N❞✐✈✐❞❡BD ✐♥t♦ t❤r❡❡ ❡q✉❛❧ ♣❛rts✳
✸✳ ❚❤❡ ☞rst ❢♦✉r ❞✐❣✐ts ♦❢ ❛ ❝❡rt❛✐♥ ♣♦s✐t✐✈❡ ✐♥t❡❣❡r n ❛r❡ ✶✶✸✼✳ Pr♦✈❡ t❤❛t
t❤❡ ❞✐❣✐ts ♦❢ n ❝❛♥ ❜❡ s❤✉✎❡❞ ✐♥ s✉❝❤ ❛ ✇❛② t❤❛t t❤❡ ♥❡✇ ♥✉♠❜❡r ✐s
❞✐✈✐s✐❜❧❡ ❜② ✼✳
✹✳ ■♥ ❛ ❣✐✈❡♥ ♣❡♥t❛❣♦♥ABCDE,tr✐❛♥❣❧❡sABC, BCD, CDE, DEA❛♥❞EAB
❛❧❧ ❤❛✈❡ t❤❡ s❛♠❡ ❛r❡❛✳ ❚❤❡ ❧✐♥❡s AC ❛♥❞ AD ✐♥t❡rs❡❝t BE❛t ♣♦✐♥ts M
❛♥❞N. Pr♦✈❡ t❤❛tBM=EN.
✺✳ Pr♦✈❡ t❤❛t t❤❡ s✉♠ ♦❢ t❤❡ sq✉❛r❡s ♦❢ t✇♦ ❝♦♥s❡❝✉t✐✈❡ ♣♦s✐t✐✈❡ ✐♥t❡❣❡rs ❝❛♥♥♦t ❜❡ ❡q✉❛❧ t♦ ❛ s✉♠ ♦❢ t❤❡ ❢♦✉rt❤ ♣♦✇❡rs ♦❢ t✇♦ ❝♦♥s❡❝✉t✐✈❡ ♣♦s✐t✐✈❡ ✐♥t❡❣❡rs✳
✻✳ ■♥ △ABC, t❤❡ s✉♠ ♦❢ t❤❡ s✐❞❡s ✐s 2s ❛♥❞ t❤❡ r❛❞✐✉s ♦❢ t❤❡ ✐♥❝✐r❝❧❡ ✐s r. ❚❤r❡❡ s❡♠✐❝✐r❝❧❡s ✇✐t❤ ❞✐❛♠❡t❡rs AB, BC ❛♥❞ CA ❛r❡ ❞r❛✇♥ ♦♥ t❤❡
♦✉ts✐❞❡ ♦❢ABC.❆ ❝✐r❝❧❡ ✇✐t❤ r❛❞✐✉stt♦✉❝❤❡s ❛❧❧ t❤r❡❡ s❡♠✐❝✐r❝❧❡s✳ Pr♦✈❡
t❤❛t
s 2 < t≤
s
2 + 1−
√
3 2
!
r.