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# Simple Geometry

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Equilateral triangle ABC is inscribed in a circle.  Point P lies on this circle's (circumference), and PB = 8.  If AB = 7 and PA < PC, find  the ordered pair  PA and PC.

( I'm reposting someone's question because I believe it should be answered.)

Jan 7, 2020

#1
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Chord               >   PB = 8

Triangle side    >   AB = 7

Angle ABD  = 30°

Let the circle's diameter be   BD          ( Point D is 180° from point B)

Diameter         >  BD = ?                BD = AB / cos(30°)           BD = 8.083

Chord              >  PD = ?             PD = sqrt [(BD)² - (PB)²]      PD = 1.155

Angle      PBD > b                 cos(b) = PB / BD                       ∠ b = 8.213°

Angle      ABP  > a                a = ∠ABD - ∠b                           ∠ a = 21.787°

Angle      PBC  > c                c = ∠ABC - ∠a                           ∠ c = 38.213°

PA = ?                   (PA)² = (AB)²+(PB)² - 2(AB)(PB) cos(a)         PA = 3

PC = ?                   (PC)² = (BC)²+(PB)² - 2(BC)(PB) cos(c)          PC = 5  Jan 7, 2020
edited by Dragan  Feb 17, 2020
#2
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Thanks Dragon,

but

next time please put a link between the original and the new questions Dragan. (one on both ends)

Jan 8, 2020