Prove Isosceles

[BMAD]

Point P is inside ABC, and is such that PAC = 18°, PCA = 57°, PAB = 24°, and PBA = 27°.

Show that ABC is isosceles.

[bonanova]

Find the unknown sides and angles:

Let

PA = 1

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

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

Central angles clockwise from CPB:

126^o

129^o

105^o

Sides:

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

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

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

Angle:

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

PCB = 39^o

PBC = 15^o