1128. Alpha. Beta. Gamma. Delta.

    The four internal angles of a quadrilateral, measured in degrees, are natural numbers and are in geometric progression: alpha, beta, gamma and delta. Calculate alpha, beta and delta when gamma is...

    a) ... 81°
    b) ... 90°
    c) ... 96°

    Pepe Chapuzas solved this problem (these three problems). He found the natural solutions and the other solutions...

    Dear Teacher:
    The sum of the four internal angles of a quadrilateral equals 360°

alpha + beta + gamma + delta = 360°

    If  R  is the common ratio of the geometric progression...

gamma/R² + gamma/R + gamma + gamma·R = 360

gamma + gamma·R + (gamma−360)·R² + gamma·R³ = 0

    Then...

    a) 

81R³ − 279R² + 81R + 81 = 0

9R³ − 31R² + 9R + 9 = 0

R = 3
alpha = 81°/9 = 9°

beta = 81°/3 = 27°

delta = 3 · 81° = 243°

    b)

90R³ − 270R² + 90R + 90 = 0

R³ − 3R² + R + 1 = 0

R = 1
alpha = beta = gamma = delta = 90°

    The progression is constant and the quadrilateral is a square. 

    c)

96R³ − 264R² + 96R + 96 = 0

4R³ − 11R² + 4R + 4 = 0

R = 2
alpha = 96°/4 = 24°

beta = 96°/2 = 48°

delta = 2 · 96° = 192°

    The other solutions are irrational (even negative).

    a)

    b) 

    c)