SOLVING PROBLEMS IN GEOMETRY: INSIGHTS AND STRATEGIES (Mathematical Olympiad Series) by HANG KIM HOO ET AL

Author:HANG KIM HOO ET AL
Language: eng
Format: azw3
ISBN: 9789814590723
Publisher: WSPC
Published: 2017-05-19T04:00:00+00:00

(2) and (3) give that AB · CD + BC · AD

= AP · BD + BD · PC

= (AP + PC) · BD ≥ AC · BD because AP + PC ≥ AC.

Notice that the equality holds if and only if P lies on AC, i.e., ∠ADB = ∠BCA and ABCD is cyclic.

Ptolemy’s Theorem is useful when solving problems regarding sides and diagonals about cyclic quadrilaterals. Refer to Example 3.1.10. One may see the conclusion immediately by applying Ptolemy’s Theorem.

Example 4.4.2 Refer to the diagram below. ABCD is a cyclic quadrilateral. Show that: sin(∠1 + ∠2) · sin(∠2 + ∠3) · sin(∠3 + ∠4) · sin(∠4 + ∠1) ≥ 4sin ∠1 · sin ∠2 · sin ∠3 · sin ∠4.

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