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Prove Isosceles

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Point P is inside triangle.gifABC, and is such that angle.gifPAC = 18°, angle.gifPCA = 57°, angle.gifPAB = 24°, and angle.gifPBA = 27°.

p136.gif

Show that triangle.gifABC is isosceles.

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Find the unknown sides and angles:

Let

PA = 1

PB = sin(24o)/sin(27o) = 0.895914

PC = sin(18o)/sin(57o) = 0.368460

Central angles clockwise from CPB:

126o

129o

105o

Sides:

AB2 = PA2 + PB2 - 2 PA PB cos(129o) = 1.71181106

BC2 = PB2 + PC2 - 2 PB PC cos(126o) = 1.151734499

CA2 = PC2 + PA2 - 2 PC PA cos(105o) = 1.151734499

Angle:

sin(PCB) = sin(126o) PB/BC = 0.62932039

PCB = 39o

PBC = 15o

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