Japan Junior Mathematical Olympiad

Question

Put 6 points A, B, C, D, E, F on the circumference of a circle in this order in such a way that the following conditions are satisfied: arc AB and arc BC, arc CD and arc DE, and arc EF and arc FA have same lengths, respectively. Determine the value of the angle BFD if ACE = 68^. (Caution: the following diagram may not be accurate.) FIGURE_0

Accepted Answer

From the well-known property of a quadrilateral inscribed in a circle, we have

∠ A C E + ∠ A F E = 18 0^{\circ}

, and therefore,

∠ A F E = 18 0^{\circ} - 6 8^{\circ} = 11 2^{\circ}

.

Since the arcs

C D

and

D E

have the same lengths, we have from the theorem on inscribed angles that

∠ C F D = ∠ D F E

, from which we conclude that

∠ C F D = \frac{1}{2} ∠ C F E

.

Similarly, we have

∠ B F C = \frac{1}{2} ∠ A F C

.

Consequently,

∠ B F D = ∠ B F C + ∠ C F D = \frac{1}{2} (∠ A F C + ∠ C F E) = \frac{1}{2} ∠ A F E = 5 6^{\circ} .