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Fun Geometry Problem with Solution #30
โจทย์
จงพิสูจน์ว่า x = 9
°
พิสูจน์
(1)
∵
AB = AC <=>
∆
ABC เป็น
∆
หน้าจั่ว ที่มี
∠A (= 90
°) เป็นมุมยอด <=>
∠ABC (=
∠ACB) = 45
° <=>
∠ABP = 45
° - 2x
(2)
∵
∠BAC = 90
° <=>
∠CAP = 90
° - 5x <=>
∠APC = 90
° + 2x
(3) ต่อ BP ออกไปยังจุด Q โดยที่ AQ = AB <=> ∆ABQ เป็น ∆หน้าจั่ว ที่มี
∠A เป็นมุมยอด <=>
∠AQB =
∠ABQ <=>
∠AQB = 45
° - 2x <=>
∠BAQ = 90
° + 4x <=>
∠CAQ = 4x
นอกจากนี้ AQ = AB <=> AQ = AC <=>
∆
ACQ เป็น
∆
หน้าจั่ว ที่มี
∠A (= 4x) เป็นมุมยอด <=>
∠AQC (=
∠ACQ) = 90
° - 2x <=>
∠AQC +
∠APC = 180
° <=>
☐
APCQ สามารถแนบในวงกลมได้ <=>
∠AQP =
∠ACP <=>
45° - 2x = 3x <=> x = 9°
Q.E.D.
Create Date : 21 กรกฎาคม 2557
Last Update : 21 กรกฎาคม 2557 0:01:02 น.
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