7日間コース
Master Circle Theorems in 7 Days
Master inscribed angle theorem, arcs, cyclic quadrilaterals, and circle-based proofs.
対象: Junior High 3 / Overseas Japanese students
Concepts & Worked Examples
Inscribed angle theorem, arcs, converse, and circle proofs.
定理： 円周角は、同じ弧に対する中心角の半分。
∠円周角 = (1/2) × ∠中心角
例：中心角 = 80° → 円周角 = 40°
同じ弧に対する円周角はすべて等しい。
直径に対する円周角 = 90°
等しい弧 → 等しい円周角（逆も成り立つ）
弧ABの長さ = 弧CDの長さ なら、円周角も等しい。
4点A, B, C, Dについて ∠ACB = ∠ADB ならば、4点は同一円周上にある（共円）。
応用：円に内接する四角形ABCDでは
In a circle, central angle ∠AOB = 100°. Find the inscribed angle ∠ACB (C on the major arc).
解答
∠ACB = 100°/2 = 50°
Inscribed angle = half of central angle on the same arc.
AB is a diameter. C is on the circle. ∠BAC = 35°. Find ∠ABC.
解答
∠ACB = 90° (semicircle) ∠ABC = 180° − 90° − 35° = 55°
Diameter gives 90° inscribed angle. Then use angle sum = 180°.
Cyclic quadrilateral ABCD. ∠A = 110°. Find ∠C.
解答
∠C = 180° − 110° = 70°
Opposite angles in a cyclic quadrilateral are supplementary.
Basic Practice
Inscribed angles, central angles, and semicircle problems.
円の角度計算を練習。
Central angle = 140°. Find the inscribed angle.
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答え
70°
Inscribed angle = 25°. Find the central angle.
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答え
50°
AB is a diameter. ∠BAC = 50°. Find ∠ACB.
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答え
90° (semicircle theorem)
Two inscribed angles subtend the same arc. One is 42°. What is the other?
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答え
42° (inscribed angles on same arc are equal)
Cyclic quadrilateral: ∠B = 85°. Find ∠D.
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答え
95° (opposite angles sum to 180°)
Standard Practice
Arc-angle relationships and guided circle proofs. ★2-3.
Pattern Practice
All circle theorem patterns. ★2-3.
Applied Practice
Multi-step circle problems with proofs. ★3-4.
Advanced Practice
Complex circle proofs combining similarity. ★4-5.
Final Test & Bridge to Next
Comprehensive test. Preview of Pythagorean theorem.